I felt good/well when my family said that they enjoyed it.

I think that I should pick well because it is an adverb that describes how I felt, is this correct?

This is a tricky one.

In this sentence, good is the word you want. It's an adjective, following the linking verb, "felt," and describing "I."

The confusing part is deciding whether feel/felt is being used as a linking or action verb.

Examples:
I feel well enough to go to school.
I feel good about my accomplishment.

Yes, you are correct. In this sentence, the adverb "well" is used to describe how you felt. When describing feelings or emotional states, we usually use adverbs (words that modify verbs, adjectives, or other adverbs) to provide more details about the action or condition being described. In this case, "well" is used to describe how you felt, so it is the appropriate word choice.