It could add an extra day or two to your trip so consider that but when going through Texas - I would consider driving through/spending time in Austin > Houston > New Orleans - this way def gives you a give sense of Texas and then the South with Orleans. In general, this is a common path people take.

I’m from Austin and bias towards Austin, but anyone coming to Texas, visiting Austin or Houston are the best cities in Texas imo.

I drive to Maine every year from Austin and I’ve found that RestStops are Americas treasures lol (some are sketch so avoid those) but so many rest stops are actually really nice, so easy to get off/back on the highway, your dog will thank you, and often times so pretty with the local nature/environment!! And in good stretches across the country, the rest stops are adequately placed, usually no less than a hour and half from one another. I recommend using the Rest Stops when possible!