Feeling Bad(ly)

Bad is an adjective, describing a noun. I am bad. My health is bad. I feel bad.

Badly is an adverb, describing a verb. I play golf badly. I slept badly last night.

If you say, "I feel badly," it suggests that you are not very good at feeling. "I feel badly because I'm wearing thick gloves" makes sense.

If you don't feel good, you feel bad. Not badly. It would be correct to say, "I feel bad because I write badly." And you should.