Now that the Lambda Layers are released you can easily share libraries and code between your Lambda Functions.

You can create a zip file for the Layer pretty much the same way as you can do so for a Function.
To share pymysql package you will need to create a Lambda Layer on base of the following function:

pymysql-bundle.zip/
  python/lib/python3.7/site-packages/pymysql

Then from your Lambda Function's code you can reference it like this:

from pymysql import ...

Little late to the game but thought it wouldn't hurt to share. Best practices suggest that one separate the handler from the Lambda's core logic. Not only is it okay to add additional definitions, it can lead to more legible code and reduce waste--e.g. multiple API calls to S3. So, although it can get out of hand, I disagree with some of those critiques to your initial question. It's effective to use your handler as a logical interface to the additional functions that will accomplish your various work. In Data Architecture & Engineering land it's often less-costly and more efficient to work in this manner. Particularly if you are building out ETL pipelines, following service-oriented architectural patterns. Admittedly, I'm a bit of a Maverick and some may find this unruly/egregious but I've gone so far as to build classes into my Lambdas for various reasons--e.g. centralized, data-lake-ish S3 buckets that accommodate a variety of file types, reduce unnecessary requests, etc...--and I stand by it. Here's an example of one of my handler files from a CDK example project I put on the hub awhile back. Hopefully it'll give you some useful ideas, or at the very least not feel alone in wanting to beef up your Lambdas.

import requests
import json
from requests.exceptions import Timeout
from requests.exceptions import HTTPError
from botocore.exceptions import ClientError
from datetime import date
import csv
import os
import boto3
import logging

logger = logging.getLogger()
logger.setLevel(logging.DEBUG)

class Asteroids:
    """Client to NASA API and execution interface to branch data processing by file type.
    Notes:
        This class doesn't look like a normal class. It is a simple example of how one might
        workaround AWS Lambda's limitations of class use in handlers. It also allows for 
        better organization of code to simplify this example. If one planned to add
        other NASA endpoints or process larger amounts of Asteroid data for both .csv and .json formats,
        asteroids_json and asteroids_csv should be modularized and divided into separate lambdas
        where stepfunction orchestration is implemented for a more comprehensive workflow.
        However, for the sake of this demo I'm keeping it lean and easy.
    """

    def execute(self, format):
        """Serves as Interface to assign class attributes and execute class methods
        Raises:
            Exception: If file format is not of .json or .csv file types.
        Notes:
            Have fun!
        """
        self.file_format=format
        self.today=date.today().strftime('%Y-%m-%d')
        # method call below used when Secrets Manager integrated. See get_secret.__doc__ for more.
        # self.api_key=get_secret('nasa_api_key')
        self.api_key=os.environ["NASA_KEY"]
        self.endpoint=f"https://api.nasa.gov/neo/rest/v1/feed?start_date={self.today}&end_date={self.today}&api_key={self.api_key}"
        self.response_object=self.nasa_client(self.endpoint)
        self.processed_response=self.process_asteroids(self.response_object)
        if self.file_format == "json":
            self.asteroids_json(self.processed_response)
        elif self.file_format == "csv":
            self.asteroids_csv(self.processed_response)
        else:
            raise Exception("FILE FORMAT NOT RECOGNIZED")
        self.write_to_s3()

    def nasa_client(self, endpoint):
        """Client component for API call to NASA endpoint.
        Args:
            endpoint (str): Parameterized url for API call.
        Raises:
            Timeout: If connection not made in 5s and/or data not retrieved in 15s.
            HTTPError & Exception: Self-explanatory
        Notes:
            See Cloudwatch logs for debugging.
        """
        try:
            response = requests.get(endpoint, timeout=(5, 15))
        except Timeout as timeout:
            print(f"NASA GET request timed out: {timeout}")
        except HTTPError as http_err:
            print(f"HTTP error occurred: {http_err}")
        except Exception as err:
            print(f'Other error occurred: {err}')
        else:
            return json.loads(response.content)

    def process_asteroids(self, payload):
        """Process old, and create new, data object with content from response.
        Args:
            payload (b'str'): Binary string of asteroid data to be processed.
        """
        near_earth_objects = payload["near_earth_objects"][f"{self.today}"]
        asteroids = []
        for neo in near_earth_objects:
            asteroid_object = {
                "id" : neo['id'],
                "name" : neo['name'],
                "hazard_potential" : neo['is_potentially_hazardous_asteroid'],
                "est_diameter_min_ft": neo['estimated_diameter']['feet']['estimated_diameter_min'],
                "est_diameter_max_ft": neo['estimated_diameter']['feet']['estimated_diameter_max'],
                "miss_distance_miles": [item['miss_distance']['miles'] for item in neo['close_approach_data']],
                "close_approach_exact_time": [item['close_approach_date_full'] for item in neo['close_approach_data']]
            }
            asteroids.append(asteroid_object)

        return asteroids

    def asteroids_json(self, payload):
        """Creates json object from payload content then writes to .json file.
        Args:
            payload (b'str'): Binary string of asteroid data to be processed.
        """
        json_file = open(f"/tmp/asteroids_{self.today}.json",'w')
        json_file.write(json.dumps(payload, indent=4))
        json_file.close()

    def asteroids_csv(self, payload):
        """Creates .csv object from payload content then writes to .csv file.
        """
        csv_file=open(f"/tmp/asteroids_{self.today}.csv",'w', newline='\n')
        fields=list(payload[0].keys())
        writer=csv.DictWriter(csv_file, fieldnames=fields)
        writer.writeheader()
        writer.writerows(payload)
        csv_file.close()

    def get_secret(self):
        """Gets secret from AWS Secrets Manager
        Notes:
            Have yet to integrate into the CDK. Leaving as example code.
        """
        secret_name = os.environ['TOKEN_SECRET_NAME']
        region_name = os.environ['REGION']
        session = boto3.session.Session()
        client = session.client(service_name='secretsmanager', region_name=region_name)
        try:
            get_secret_value_response = client.get_secret_value(SecretId=secret_name)
        except ClientError as e:
            raise e
        else:
            if 'SecretString' in get_secret_value_response:
                secret = get_secret_value_response['SecretString']
            else:
                secret = b64decode(get_secret_value_response['SecretBinary'])
        return secret

    def write_to_s3(self):
        """Uploads both .json and .csv files to s3
        """
        s3 = boto3.client('s3')
        s3.upload_file(f"/tmp/asteroids_{self.today}.{self.file_format}", os.environ['S3_BUCKET'], f"asteroid_data/asteroids_{self.today}.{self.file_format}")


def handler(event, context):
    """Instantiates class and triggers execution method.
    Args:
        event (dict): Lists a custom dict that determines interface control flow--i.e. `csv` or `json`.
        context (obj): Provides methods and properties that contain invocation, function and
            execution environment information. 
            *Not used herein.
    """
    asteroids = Asteroids()
    asteroids.execute(event)

You can define a service class that is separate from the handler. Based on the event, you can invoke different methods in service class. You can also extend this idea more via Factory pattern.

lambda(event) {
   ...inspect event
   ...instantiate service class
   ...use service class methods to achieve goal
}

Putting the expressions in a list may simulate multiple expressions:

E.g.:

lambda x: [f1(x), f2(x), f3(x), x+1]

This will not work with statements.

You can have multiple functions in a single class. It's just that you have to set the required function as a handler for a particular API gateway on AWS which you are using it for the lambda function that you created. Then when we run our lambda it will only call the set handler function leaving the rest of the functions untouched.

You can even use rest of the function to work with other lambdas or even as a normal function which is called by the handler function internally.

You need to go through the aws lambda documentation for more information.