3spd auto not cooperating in a 65

Actually the car is an early 65 but it doesn't really matter, A C4 is a C4. My problem is that I put it in Drive and it pulls 1st and 2nd fine, but I never feel it slip into 3rd. If I pull it into 2nd it takes off and stays in 2nd same thing with 1st gear. I've tried playing with the shifter a little bit but since it takes off in 1st and shifts into 2nd. I assume I'm looking at something deeper.

Has anyone on here dealt with this problem without taking it to a tranny shop? Looking to save some money, since I got a few tools laying around but if I gotta pull the tranny I'd better off to take it elsewhere. Thoughts anything would be helpful, Thanks in advance.
 
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If the car isn't driven regularly, that's going to happen. Try adding a bottle of Lucas "transmission treatment" to it and see if it helps. The stuff frees up sticky parts in the transmission. If it doesn't, you've got seals that have dried and shrunk from inactivity and are letting the fluid bypass.