Ok I movied into this new apartment in August when I noticed that the air conditioner would run OK but then I noticed that it wouldn't cut on/off by itself which is a major concern while I am sleep,at work or school. I have to move the thermostat up and down from 90degrees to 50degrees a few times before it will shut off or which it from cool to off to heat back and forth a few times. I told the complex and they installed a new thermostat but it is still doing the same thing. Two weeks ago my air was blowing out heat when it should've been cold and maintenance guy said my "capacitor" had blowed. Now since being here I have had 3 outlandish electric bills with last months' almost $200 for a freaking 1 bedroom. If I tell the complex again this will be their 4th trip out to my unit and I'm starting to get pissed off. Does anyone know what the problem is and if so is it wrong of me to call a professional a/c guy and have them bill my complex? Thanks
btw:the unit inside and fan outside both stay on or off together and I have a white - rodgers thermostat and know nothing about wiring

You failed to mention if it is a heat pump or not so I will assume that it is not. First thing I would check would be the control wiring. If any of the conductors have become exposed and touch it can cause this problem.