I have a water-logged meat thermometer (non-digital) How do I dry it out?
If water got in you can get it out. Your thermometer is the circular analog type and it has more than one piece. There's one piece with the probe and dial, and then there's a circular cover with the glass on it. The water got into the thermometer through the seam where the 2 pieces meet, and this is where you'll get the water out, you just have to figure out how. 

What I would do first is to get a safety pin and see if I can pry the gap between the 2 pieces apart a bit to let the water run out. Be careful not to poke yourself if you try that though, it's easy to do. If you do get it dripping out then you'll need to find a way to hold it in place at about a 45 degree angle from vertical in the direction of the pin to allow the water to run out. Sticking it in a piece of styrofoam or a potato should do the trick. 

Another option would be to put it in a warm (call it 150-160F, 70C) oven at about a 45 degree angle from vertical. Sticking it through a medium size potato will keep it stable, don't use styrofoam as the fumes will be toxic. 

You could also combine these 2 methods. 

Another option I would consider is to put the thermometer on its side in a sealed container full of uncooked rice. The rice absorbs water and makes the air very dry, it's a method which works to dry out watches and even mobile phones.