Welcome to the forum, it is a pretty neat place to hang out at. As for your question about varmints, it is easier to go with the list of what is considered a game animal in Texas.
Animals that there is an actual season on are:
White-tail and Mule Deer.
Pronghorn.
Desert Bighorn Sheep.
Javelina.
Red and Grey Sqirrels.
There is no open season on Black Bear, and there are a few in Texas, not many, but their numbers are growing.
Mountain Lions at this time are given no protection, T.P.& W. just wants to know if you kill one, where did you kill it at, that includes if you run over one. They don't have a real good idea the size of the population of lions in the state, so they are just wanting to find out where there are or maybe lions. At some point they may classify them as a game animal.
Mink, Otter, Badger, Ringtail, Red and Grey fox are listed as fur bearers, but as long as you aren't trying to sell the pelts, there is nothing wrong with taking any of them if you see them.
Coyote are wide open, and can be shot anytime you are on a place where you have permission to hunt.
Bobcats have special rules if you are going to try and sell the pelts. You have to get a CITES permit.
You will have to have a hunting license for any hunting you do in the state.
Oh, I forgot to mention the biggest varmints of all Feral Hogs. They ain't really varmints, but they are a lot of fun to hunt and they taste pretty good also.
Again Welcome to the forum, hope you enjoy it.
