+1 for glove liners with a note of caution – if they make the overall fit tight you may find it worse than without the liner.
It could be your position on the bars. Have you tried changing your “grip” around as you ride. I went out with my club in the bitter cold before X-mas. On one part of the ride my fingers were cold and my thumbs were fine, then later fingers good, thumbs bad. I find that if riding on quiet roads and not racing a good position to minimise hand pressure is to ride with the fleshy part of your palm in contact with the bar and fingers and thumb almost making a loose fist so your thumb joint isn’t taking any pressure. Not sure How well I have described that, hope you understand what I mean.
Some of the coldness felt in extremities when cycling is down to pressure and restricted blood flow.