It's just fine to give your snake a wet mouse, just wipe it with a cloth or paper towel to dry it up a bit. My snakes don't mind eating a damp mouse, just as long as they get to eat! Lol. It's totally fine to feed your snake a wet mouse though, they don't seem to mind it much.

Also, instead of waiting hours for a thawed out mouse, I suggest putting the mouse in a container of really hot water. Let it thaw in the water for about 5-10 minutes, maybe longer if the mouse still needs to thaw a little longer (15 min). Right when you take out the mouse, dry it off with a cloth or paper towel, then offer it to the snake while still warm. If the snake doesn't take it after a minute or so and the mouse feels a little cool then dip the mouse in the water for 30-60 seconds.