If you want to drive in snow the summer tires are not proper tool. In wet snow some summer tire may be passable but do not count on it. I was caught with Hankook Evo V12 in wet october snow last year and they performed better than expected. But my expectations were quite low.
If you have dry snow accumulating then take your scooter, may be safer. Or stay home.
If you have to drive in the snow in Texas and have money use summer tires on dedicated wheels for most time and put studless winter tires for snow/ice on dedicated wheels (I think you have more chance of glare ice than snow but I do not know your wether). You may want to put them on only for winter weather. With TX wether you almost need a special vehicle to handle winter as it may be that sudden and infrequent.
Cheap solution is to buy all seasons but on ice they will not be of great help.
Decent all seasons are Goodyear Eagle GT, Conti DWS but you do not need their very good snow traction that often, Potenza RE970.
Maybe Falken ZE-912?
PS Looking at tire prices in your selected size. Maybe go with smaller wheels if your main objective is: "Just want something that works well in the rain and decent in the snow/winter". Narrower and smaller will buy you decent all seasons for much less money and be better with hydroplaning and snow.