Different American anon here, and also kinda drunk and kinda trained on reading non-American English lit. I fucking love this fic to all hell, so I'm gonna chime in on Americanizing the English. California born and bred so I can't speak for New York slang but hey general American!English stuff:

"go back to my place" is also valid

I think some of the British "yeah?" parentheticals can be replaced with "ya know?" or "right?". So like "Just.... do that again, ya know, with the teeth." or "this is okay, right?"

"Bit desperate, were you" could just be "Bit desperate, huh?"

"I swear I’m not taking the piss" can be Americanized as "I swear I'm not fucking with you"

Unfortunately we do not say "wank". The common substitutes are "jack off" and "jerk off"