"Well" does not serve any grammatical usage in this sentence. That's because, in this context, "well" is an interjection, a word that expresses emotion. Depending on how it is enunciated, "well" could indicate impatience, surprise, nervousness, and a variety of other emotions.

However, here it seems to function as a filler, similar to "uh." It doesn't have any true meaning. People type it for the same reason as they type the interjections "oh" or "uh" - to convey emotion or to fill space while thinking of what to say.

Answer from Grace on Stack Exchange
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Cambridge Dictionary
dictionary.cambridge.org › us › grammar › british-grammar › well
Well - English Grammar Today - Cambridge Dictionary
Fiona is feeling better. Well, she’s much better than she was. She’ll be back to work on Monday. We can use well when we admit or acknowledge that something is correct or true:
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What is a synonym for well?
Some synonyms and near synonyms of well(adverb) include: · Ably · Acceptably · Admirably · Competently · Deftly · Effectively · Efficiently · Satisfactorily · Try Scribbr’s Paraphraser to find more alternatives for words like “well.”
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scribbr.com
scribbr.com › home › good vs. well | difference, definitions & examples
Good vs. Well | Difference, Definitions & Examples
What is a synonym for good?
Some synonyms and near synonyms of good (adjective) include: · Acceptable · Excellent · Favorable · Great · Marvelous · Praiseworthy · Satisfactory · Try our Paraphraser to find even more alternatives for “good.”
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scribbr.com
scribbr.com › home › good vs. well | difference, definitions & examples
Good vs. Well | Difference, Definitions & Examples
Is it “going good” or “going well”?
Going well is a phrase meaning “proceeding in a satisfactory way.” People sometimes use “going good” to mean the same thing, but this is technically incorrect. While “good” is sometimes used as an adverb in everyday speech, this should be avoided in formal contexts. · This is also the case for doing well and “doing good.”
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scribbr.com
scribbr.com › home › good vs. well | difference, definitions & examples
Good vs. Well | Difference, Definitions & Examples
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Scribbr
scribbr.com › home › good vs. well | difference, definitions & examples
Good vs. Well | Difference, Definitions & Examples
March 11, 2023 - Well and good is part of the expression “that’s all well and good.” It’s used in contrast with a following clause to say that while things may seem fine, there may be unexpected consequences or unresolved issues. In this context, both words are used as adjectives. Examples: Well and good in a sentencePerson A: I have a dentist appointment tomorrow, so you’ll have to present the report to the committee.
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Merriam-Webster
merriam-webster.com › dictionary › well
WELL Definition & Meaning - Merriam-Webster
Both well and good are adjectives when they follow feel to express a meaning of good health, and both are grammatically correct. In "I feel good" there is often a connotation of good spirits in addition to good health, but the two phrases are ...
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Quora
quora.com › Why-do-people-use-the-word-well-when-starting-a-sentence
Why do people use the word “well” when starting a sentence? - Quora
Beginning a sentence with well can indicate that you are getting ready to answer a question, but are not quite done thinking through your answer. Or it can indicate a change in the course of the conversation.
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YourDictionary
sentence.yourdictionary.com › home › well
Examples of "Well" in a Sentence | YourDictionary.com
Well, the Internet is bigger than air conditioning. ... Well, then, let's be off. ... Father will need to rest as well.
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Merriam-Webster
merriam-webster.com › sentences › well
Examples of 'WELL' in a Sentence | Merriam-Webster
Skiers and snowboarders should also watch for tree wells in the deep snow, the resort said. — Austindedios, oregonlive, 1 Apr. 2023 · At the time, officials believed it could be related to a well drilled at the site. — Pilar Arias, Fox News, 10 Sep.
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Collins Dictionary
collinsdictionary.com › sentences › english › well
Examples of 'WELL' in a sentence | Collins English Sentences
Mix all the ingredients well. Wash your hands well with soap. 'He speaks well of you.'—'I'm glad to hear that.'
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Microsoft
microsoft.com › home › writing › ‘good’ vs. ‘well’: when to use which
'Good' vs. 'Well': When to use which
August 27, 2024 - I did well on the test. In the sentence above, “well” modifies the verb “did.”
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Reddit
reddit.com › r/grammar › what is the correct way to use "good" vs "well" in this sentence?
r/grammar on Reddit: What is the correct way to use "good" vs "well" in this sentence?
January 22, 2022 -

From my recollection, "good" is an adjective and "well" is an adverb. I recall adjectives modifying/ qualifying nouns and adverbs modifying/qualifying verbs.

If this is correct, why do people say "I am well"? Is there a rule I am forgetting?

I was thinking that if we are looking to qualify the subject "I", then we could satisfy that with either "good" or "well".

If we chose "good", we would say "I am good" .

This would be consistent with other examples where the adjective is at the end of the sentence (e.g. "I am happy.")

If we chose "well", we would say "I am doing well" .

This would be consistent with other examples where a present tense verb preceeds the adverb.

(e.g. "I am happy.")

Unless I am unaware of some rule, I can't see how it is grammatically correct way to say "I am well".

Hopefully someone can clarify the rules for me.

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Grammarly
grammarly.com › blog › commonly-confused-words › good-well
Good vs. Well–How Should I Use Them? | Grammarly
December 23, 2020 - All you need to remember when you ... for your sentence is that good modifies a person, place, or thing, whereas well modifies an action. If you’re having a good day, then your day is going well. Here’s a tip:You don’t have to guess whether you’re using certain words correctly or breaking grammar rules in your ...
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Study.com
study.com › english courses › english grammar rules
Difference Between Well & Good | When to Use & Examples - Lesson | Study.com
April 23, 2020 - When using linking verbs or sense verbs like "is," "look," "sound," "smell," and others like them, use "good" instead of "well." Consider: The band sounds good tonight! "Good" technically modifies "sounds," since it describes how the band sounds. However, "sounds" is a linking verb in this sentence.
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Basic English Speaking
basicenglishspeaking.com › home › “well” in a sentence (with audio)
"Well" in a Sentence (with Audio) - Basic English Speaking
July 21, 2020 - Examples of how to use the word “well” in a sentence. How to connect “well” with other words to make correct English sentences. well (adv, adj, exclamation): in a good way, to a high or satisfactory standard; healthy; not ill; used to introduce something you are going to say, often ...
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Sentence Dictionary
sentencedict.com › well.html
Well in a sentence (esp. good sentence like quote, proverb...)
1. in good health especially after having suffered illness or injury 2. resulting favorably 3. wise or advantageous and hence advisable. adv. 1. (often used as a combining form) in a good or proper or satisfactory manner or to a high standard (`good' is a nonstandard dialectal variant for `well') 2.
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Thesaurus.com
thesaurus.com › e › grammar › good-vs-well
Good vs. Well: The Grammatical Difference | Thesaurus.com
July 17, 2023 - When used to describe health, the verbs look and feel can cause some confusion about whether to pair them with good or well. In standard language, the adjective well is generally preferred when talking about physical health, and the adjective ...
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Bab.la
en.bab.la › sentences › english › well
well example sentences - Use well in a sentence
English It was well organised and it is also a contribution to better communication, and to the engagement of young people as we call for future EU policies. volume_up more_vert ... English The Court is well aware of its need, in a constantly changing Union, for continuing self-improvement and for the best possible use of its resources.
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Cambridge Dictionary
dictionary.cambridge.org › us › dictionary › english › well
WELL | definition in the Cambridge English Dictionary
... Invite Emily - and Scott as well. See more · WellWell is a discourse marker, adverb or adjective. … · Well as a discourse markerIn speaking, we often use well at the start of what we say.
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YourDictionary
yourdictionary.com › vocabulary › versus › meanings › when to use good vs. well for correct grammar
When to Use Good vs. Well for Correct Grammar | YourDictionary
March 25, 2021 - The house smells good. She looks good in that dress. The car appears good on the exterior. The idea seems good to me. When using linking verbs that describe health such as be, feel or look, use "well."