Is badly an adverb

Yes the Word badly is an adverb.
Badly is almost always used as an adverb.

Examples:

She runs badly. (Badly is describing the way she runs.)

He wants a new TV badly. (Badly is describing the how much he wants the TV. Wants is the verb.)

They did badly in the elections. (They did not do well.)

However, it may be considered an adjective when it is used with the verb "feel."

"He felt badly about the mistake."

Here, the opposite would be 'he felt good' about it, and good is an adjective. On the other hand, referring to health, saying 'he felt good' is actually an informal way of saying 'he felt well.'