I was also going to suggest Gabapentin or Lyrica. You could also get a compounding pharmacy to make up a topical cream with Gabapentin, Amitriptyline, Lidocaine, and an anti inflammatory such as Naproxen. Some pharmacies have experience with making up their own cocktail for neuropathy/ neurogenic pain.
There is also topical CBD ointment, but it would be trial and error, just like with the other suggestions.
Then there’s always Ropinirole.
Best of luck with this, whatever you end up doing.
My symptoms may indeed represent a different process, but FWIW, I don’t have the sensation of an extremity “going to sleep”.