You can shoot your shot with the FAA. And make 50k in 6 months..... or you can live wherever, getting paid *typically* significantly less. As for leases, there's plenty that do month to month. And as long as you're not renting from a agency, typically owners will let you out early if you have a "I got a new job across the country, can we work something out?" Attitude.
All of this isn't even counting the networking/experience you gain from actually working at a new tower. Sitting on your hands or doing anything outside of air traffic simply because you don't wanna move is absolutely silly.