Do you leave the oven thermometer inside the oven while you are baking? We have a very old gas oven at home. I just started baking but every time I use it for baking, I end up having burnt cookies, unevenly baked cakes, etc. 
I plan on buying an oven thermometer. I'd like to know if I should leave the thermometer inside the oven in order to  monitor the temperature. Or should I take it out once the oven has been preheated enough?
In your case it's definitely worth leaving it in, because it will help you get to the bottom of whether things are burning because the temperature keeps rising during cooking.

Oven thermometers are generally designed so that they can be left in. I prefer the type that can hang from a rack