That's a cool find! I don't have my books handy so I don't know if the Texas DPS used 4 door Plymouths in 1962. They had 2 door sedans in 1960 and 1961. I have a 1961 Pursuit, ex TX DPS Highway Patrol. 361 4bbl and three speed stick like yours. Also, dual rear antennas positioned like on your car with a similar antenna base. Mine had a single rotator on the roof. Those fender mounted lights look odd. You might want to join the copcar.com forum, a wealth of info there. But as they say, "keep your seatbelt fastened", the exchange of words there can sometimes be a bit tough.

Don't lose those wheels and hubcaps btw, they were standard on pursuits, optional on Patrollers, Taxis and export vehicles. Of course, if you want to sell I need a set for my car...