My toursit visa will expire next month while I am waiting for an EEA family member residence card, does this make me an overstayer?
As long as you can prove you are waiting for a decision you will not be penalized as an overstayer.
I think you have to show your case number, but am only recalling from memory.
Whatever you do, don't leave the country you may not get back in.
If the decision comes back unfavourably, you will need to leave as soon as practical.