Treating sign as spoken language makes sense and is generally the right way to go. But remember, you're not actually quoting what someone signs. You're quoting the English translation of what someone signs. So you should treat ASL signers exactly like you would treat any other foreign language.
ASL is a language in its own right, not derived from English, and has its own vocabulary, syntax, etc. It's not just finger signs but involves movements of the hands, arms, head, face, mouth, etc. E.g.,
"Where are you going?" he signed would literally be
"[YOU] [GO] [WHERE?]" he signed, substituting the proper signs for the bracketed transcriptions and using standard ASL syntax. This of course it not very helpful way to transcribe! It would be as if we transcribed French
"Où vas-tu?" as
"ooh vah too" or
"Where go you?"
The point is that ASL is a language and you should treat it as such in writing. So if you would just quote a French person speaking in French, do the same with ASL. If you italicize foreign speech, then do that. In my opinion you need to aim for two goals: (1) accurate depiction of ASL as a foreign language and (2) reader comprehension.