What would make two fingers on my non dominant hand tingle or feel as if they have been asleep?

Nerve entrapment. It may depend on the fingers in question. If it is the 4th and 5th fingers and perhaps 1/2 of the middle finger, then you may have entrapment of a nerve at the elbow or are stretching that nerve. If it is the thumb and index finger and long finger, you may have carpal tunnel syndrome. Otherwise i don't know. You would need to see your pcp or a neurologist or orthopedist to exam and diagnose.
Pinched nerve ? Carpal tunnel syndrome is very common. Other possibilities are ulnar nerve at elbow, pinched nerve at spine. See a neurologist you may need emg, x ray etc.
Carpal Tunnel Syn. Most likely carpal tunnel syndrome. You're probably thinking, "this is my non-dominant hand; therefore, it can't be carpal tunnel." that's a very common expression from patients, but I do see a lot of cases of non-dominant hand carpal tunnel !