This sounds like Lumbrical Shift Syndrome.
Probably a strain or partial tear (although I think you would have noticed the latter) of the small muscle inside the hand.
I had this a few times at the beginning of my climbing career.
The good news is, if you tape the two fingers together you should be able to climb without pain and further aggravating the injury pretty soon.
At least that's how it was for me.
Just make sure to give it a few days of rest first, maybe ice baths to facilitate circulation and lots of food and sleep.
Hope you'll be back soon!