Objectionably is the adverb from objectionable: in a way that is unpleasant and likely to offend. ‘He spoke objectionably about her family.’ It is correct English, but it is not a word most people reach for in speech; they would say offensively, or rewrite the sentence (‘what he said was objectionable’). Same formal, slightly stiff feel as the adjective. You will meet it more in writing than in talk. Use it when you truly need an adverb, and do not feel you must — English is happy to put the idea on the adjective instead.