Because your symptoms are so localized I wonder if a topical product would help you?

1) Biofreeze applied on the tops of your hands might work. You can get this now on Amazon...I'd do the gel and not the roll on, as it is more concentrated.

2) Or the new pain gel by Salonpas... Salonpas Deep Reliving Gel...this is very strong and has a salicylate in it to penetrate and exert anti-inflammatory action.

3) A Lidoderm patch (cut to fit) applied to the back of the hand might work too.

4) Some posters here get an RX compounded transdermal gel with ketamine, gabapentin, lidocaine, and sometimes an NSAID to rub into specific target areas. Yours would be the backs of the hands (the palm skin may be too thick). But you could apply on the inner wrist where the carpal ligament is.

I have found topical agents like these work very well for me and allow me to avoid major drug treatments. (My only RX is a little tramadol at night on really bad days.)

Do you wear support splints to bed? This may help too.
Your issue may be a tendonitis, or a gout thing in your hands.
Ask to have a uric acid level drawn to see if you are elevated.

I am having some urate deposits in two fingers now, but the Salonpas is helping to move them out lately. These deposit along tendons and make whitish hard lumps which may or may not show, but sometimes you can feel them. I knew a fellow once who had huge amounts of urate along his tendons, esp after an illness where his white cell count would elevate. When the cold, or infection was over, these would be removed by the body and that releases lots of purine peptides into the system and some people lack an enzyme (xanthine oxidase) to metabolize them away...so this ends up along tendons. These can be pretty painful, burn and sting.