Don't know what your financial picture looks like but, it may be more feasible to buy than rent a house. You usually pay about the same or less
with a mortgage rather than rent. Rentals can be vary expensive depending on the area. As to renting with pets, my daughter was in that same situation
with one dog ( a pit mix) and was turned down everywhere until she got a letter from her Vet (our family Vet for years) attesting to the dog's good nature.
It worked for her.