Solecism

As pointed out in @Edwin's comment referring to this earlier Answer, the relevant term here is...

Enallage
...related to solecism (a deviation from conventional word order). Enallage, however, is usually regarded as a deliberate stylistic device, whereas a solecism is commonly treated as an error of usage.

A few famous examples from advertising / TV / movies...

Got milk?
Eat fresh
We was robbed
Mistah Kurtz—he dead
Thunderbirds are go

"All of these stick in our minds because they’re just wrong—wrong enough to be right" (Mark Forsyth, Rhetorical Reasons That Slogans Stick, The New York Times, November 13, 2014)

Solecism (in the sense “Error in the use of language”), a rather general term that encompasses grammar mistakes, is one possibility. Another general term is infelicity. A few dozen quite-specific kinds of language errors (mostly stylistic rather than grammatical) are explained in BYU's Stylistic Vices webpage, for example pleonasm, “Use of more words than is necessary semantically. Rhetorical repetition that is grammatically superfluous” or acyrologia, “An incorrect use of words, especially the use of words that sound alike but are far in meaning from the speaker's intentions” (like malapropism, eggcorns, etc) and many more.

For verbs, consider misspeak (“To fail to pronounce, utter, or speak correctly”). A wikipedia article called Misspeaking says

Misspeaking is a word used to describe the act of speaking “incorrectly, unclearly, or misleadingly”, to “fail to convey the meaning one intends by one’s words”

Also consider the form to err grammatically. Verb err means to make a mistake; to err grammatically is to make a mistake in grammar.