You will need to go on the Yellowstone site and check lodging options as there are quite a few. A cabin might work well for you but "cheap lodgings" and Yellowstone don't really go together. You may get some deals at this time of year in the town of West Yellowstone , it's a case of searching. You have already had answers to some of your questions above which you did not reply to, so please keep all comments and questions here. Badlands and RMNP along with Yellowstone and the Tetons will fill your time so it's a case of quickest routes in between really.