My apartment is the same way as yours. As soon as I turn off the AC the heat and humidity almost immediately comes back. I can only turn it off for a few minutes. I set it to "energy saver" where it shuts off at 75 degrees.

Like Donna, our AC never gets into our bedroom. And per our lease, we can't put one in there.

You should be able to apply for extra help with your electric bill. I don't know exactly how to do it tho. If you have a case manager you can ask, that would be best. If not, try going to the electric company's website.