A temporary tag is issued so the dealership has the time to get your title and registration. And in Texas, it is illegal to issue a second temporary tag. Actually it is illegal in most if not all states to issue a second temporary tag. And yes, you can get a ticket, and your car can be impounded if you are driving it around with an expired temp. A temp tag is a temporary registration and if it is expired, it is just like driving with an expired registration!!
You still have to make your payments, or you will have more trouble on your hands and the bank can repo the car if not paid!!
Unfortunately, you are out of state, so it is probably inconvenient for you to go back to the dealership personally. But, if at all possible, I would go back there in person to find out what the problem is. Be calm and firm and demand a rental car until they can provide you with a proper, permanent tag for your car. Don't accept another temporary tag, it is illegal. Don't accept their word that they will take care of any tickets, because they won't. And they certainly won't pay any impound fees and storage fees if your car is impounded! Don't accept a letter from them saying there is a problem with you motor vehicle, but they are working on it (or something to that effect), it won't matter to the police if they stop you. And only if you are willing to carry through your threats, you can contact the DMV in their state and report them. Actually you can report them to the state licensing department that issued their dealer license. You said what state you are in, but didn't say what state they are in. If you paid them the tax and the licensing fees, they are required to get your tags. And the Dealer licensing board can fine and the states DMV can revoke their motor vehicle priveledges for not following state requirements. And no dealership wants their motor vehicle priviledges revoked or suspended.
You can actually contact the bank where your loan is at and tell them about the titling problem. They want their interest protected too!! They want their lien on the title, so they may be able to help you out also!! Use every tool at your disposal to get this straightened out as soon as possible.
Worked for car dealerships for 20 years.