I personally lucked out on housing here, so i'm not really sure what it's like to look for a place in town. Your best bet would probably be to find some people who are sharing a house and looking for an extra roommate. Otherwise, you can work in Teton Village at a hotel or the resort and try to get employee housing. I know that The Hostel and Hotel Terra offer cheap, almost free housing for employees.
Jackson is only expensive if you don't know how to avoid the tourist traps and expensive stores. People exaggerate it a little bit. The gas is a normal price and certain grocery stores are super cheap. The season pass is crazy pricy, however. I would recommend getting an afternoon job at the resort for the free pass. I'm living off minimum wage here with no help from family, so it is completely do-able if you live frugal.
I highly recommend trying everything you can to get here. Let me know if you have more questions!