Is it legal to buy a travel trailer and decide to sell it without transferring it into my name, just sell it AND PUT NEW BUYER ON TITLE IN TEXAS

You bought it but never titled it in your name? Then you are not the registered owner and cannot sell the trailer.

You have to put it in your name before YOU can sell it.

Obviously you cannot sign someone else's name as seller.

Yes, you will need to title it into your name, and then sell it.