Why is it so hard to rent?

I recently moved to Houston with my husband. He is currently on a Fulbright fellowship as a teacher for four months and he has already an offer as a visiting professor lined up for a three year period. We were initially very excited about this opportunity, but we’re having trouble finding a place to rent and we need help navigating this awful system. We want to stay under a 1500 budget, but it’s proving to be impossible because our visa was initially granted for the four month period (we’re changing sponsors after that) and the short term leases change dramatically. In our country you don’t have to pay an application fee, so we’re concerned about losing money that we need to make our move. Do you have any suggestions? Tricks that worked for you ?