Congratulations on being able to get plates for your vintage Airstreams! I too believe that there is inconsistent applications of the new Texas Vehicle Registration laws.
I have noticed that dual axel vehicles must go thru the hoops of inspection then presenting the paperwork to the county clerk tax folks to get your plates. At that time they want some sort of liability insurance document as well, at least for the TV. Single axel trailers under 3000 lbs. capacity are exempt, but still have to buy a plate.
Once that has been done one time, from then on you get the yearly inspection and then go online and finish your registration, pay your money and receive your sticker in the mail. The vehicle inspection sticker and the license plate in one item, not two.
All that said, the satellite office for registration in the county could care less and just takes the money and gives you your license plate sticker. The office at the court house is a different matter all together.
At our court house the property tax folks are in the same room as the vehicle tax folks. They are really nice people but they have a thing for billfolds and purses. They want yours!