There are some controversies about the word usage in this context. So, which example is grammatically correct: "I'm dressed warmly" or "I'm dressed warm"?
Both "dressed warmly" and "dressed warm" are correct and commonly used. Since dress here is a verb, you need an adverb to modify it. "Warmly" is the more common adverb, although in informal language, "warm" is also used adverbially in a restricted range of contexts.
Another example would be "wrong". Compare:
You spelled that word wrong/wrongly.
Be sure to dress warmly for skiing. [written style]
They were dressed warmly but still felt cold. [written style]
"Hey, don't forget to dress warm when you come up North." [colloquial, spoken]
"My advice? Dress warm and don't forget your gloves!"
Both are used and both are controversial. In a grammar-critical situation like an English test or formal letter, I would avoid both of them and rephrase the sentence.
The purported problem with the adjective "warm" is that there isn't a noun to be modified. One could argue that "warm" modifies "I" but it's arguable whether this is accepted usage. The purported problem with using the adverb "warmly" to modify "dressed" is that "warmly" isn't a quality of the action "dressed" but rather a quality of the unnamed clothes I'm dressed in.
I know it's freezing outside but I'm dressed to stay warm.