Do you have any fields around where you live that are surrounded or bordered by trees? Here in KY, coyotes always stick close to those areas, mostly farm land. I suggest you go to a nearby field with a .223 and have a stake out. I'm not sure what Texas Fish and Wildlife has as far as regs, but in KY, you can kill as many coyotes as you want, year-round.
It won't just be cats and dogs, if they get comfy enough, coyotes will go after infants and even small kids. There is also the ever-present danger of rabies. If you trap it, get fish and game or animal control to come out and get it, releasing it a couple counties over will do nothing. You can always use repellent (Deer-off, basically mace), but that won't stop a determined coyote. The best option is to try to find the burrow. Look at any place close that has rocky ledges or cliffs, riverbanks, holes under stumps, etc. These are prime real-estate for the ugly buggers.
Good luck.
P.s. don't mistake a fox for a coyote, if you kill a fox, your arse is in a world of hurt if you don't have tags or if it is out of season, or both.