I am no dr but it could be coming from a pinched nerve in your neck or it could it be carpal tunnel or arthritis?

Soak your hand in some lukewarm water for 5 to 10 mins to get some cirulation back in your hand ( for just right now)
if you still have pain coming in you need to see a hand doctor, cause it might be arthiritus in the palm of your hand not your fingers.

I have had two forms of hand pain, one is when you wake up in the night to hands that have 'pins & needles' & later pain...Those were my symptoms for carpel tunnel which I had surgery for last year.
Then three months ago I had my two thumbs become very painful in the base joint & I could not bend my thumbs & my little finger which when I closed my hand would stay closed. I would have to pull the finger back.
This is called trigger fingers which I had surgery for to release them.
Although that was sucessful my both hands are still swollen & very painful.
Both these conditions suddenly came when I was trying to get off the drug prednisolone which I had been taking for six years for polymyalgia
rheumatica. This drug masks all sorts of conditions & while I am off it now
i am really suffering with pain in all my joints, which I am told is inflammation. Pmr is inflammation of the muscles.