Stand-offishly is the adverb: it tells you the way someone did something — coldly, keeping their distance, as if they did not want to be close to you. "He nodded stand-offishly and walked away." The nod happened; the coldness is how it happened.

Because it is an adverb, it sits next to a verb, not next to a noun. You say a person is stand-offish, but that a person behaves stand-offishly, speaks stand-offishly, smiles stand-offishly. That last one is a nice example of what the word does: even a smile can be given in a way that keeps you outside.

I should be straight with you about how often this is used. It is rare. Native speakers write "stand-offish" many times more often than "stand-offishly", partly because it is long and a little awkward in the mouth. Most people would say "he was stand-offish about it" or "he was cool with me" rather than reach for the adverb. You will meet it in novels more than in speech.

So: know it when you read it, and feel free to use it in writing, but do not feel you must. Both spellings, stand-offishly and standoffishly, are accepted.