Unless you are going to be staying or visiting one of the smaller towns around Houston, just remember that most of the major cities in the south aren't too much different in daily stuff than the north.
And, as a Southern Baptist I can tell you even dressing for church these days can be more relaxed. For the funeral just a nice dress or slacks can be the norm. Take both in case and don't worry.
As someone else said, too if you just wear a nice and comfortable hat for you

most will not probably say anything. And, if they do just let them know the truth. You may have a new set of very good friends that way! lol
We middle aged ladies especially know about that hair loss of some sort anyway. lol You just don't find many of us down south anymore wearing hats due to hat hair and the humidity! rofl
Of course do be prepared for lots of casseroles, fried foods, etc. Sure we have the same fast food places and ethnics restaurants, but the good 'ole Southern cooking you find at a gathering after a funeral will be the 'good stuff'. lol!
Enjoy your trip!
Kim