badmouth vs sully

badmouth and sully both are verbs.

Word	Noun	Adjective	Verb	Adverb
badmouth	No	No	Yes	No
sully	No	No	Yes	No

As verbs, sully is a hypernym of badmouth; that is, sully is a word with a broader meaning than badmouth:

badmouth: speak unfavorably about
sully: charge falsely or with malicious intent; attack the good name and reputation of someone

Other hypernyms of badmouth include asperse, besmirch, calumniate, defame, denigrate, slander, smear, smirch.

badmouth (verb)	sully (verb)
speak unfavorably about	charge falsely or with malicious intent; attack the good name and reputation of someone
	make dirty or spotty, as by exposure to air; also used metaphorically
	place under suspicion or cast doubt upon