Seeking pet friendly 1bdrm STX

Hi there! My boyfriend and I are looking for a place to live October 1. We need a pet friendly place as we have a cat and will eventually get a puppy. We're looking for a home in the rain belt, where the gardens are lush, because I'd like to have a yard to garden. $500-$700 utilities included range. We are open to care taking - lots of handy man/mechanic skills as well as gardening/cleaning/pet sitting experience. Thanks so much for reading. Let me know what you've got!