The transmission you have is a Hydramatic, or, the Slim Jim Rotohydramatic. They are pieces of junk!
I have one in my 64 98 and it is DEFINITELY the weakest link in my car.
Like 88 Coupe suggested you should tune up your car before you decide to do anything expensive and difficult (yes you can put a modern tranny in your car). There is a mechanical throttle valve linkage with the Hydramatic and it is difficult to get right. To set it you have to make sure the throttle valve on the tranny is in the zero position at the same time your carburetor is in the zero position. When you hook up the throttle valve to the carb you take the adjustment out until there is no slop in the throttle valve linkage.
I know it sounds difficult but if you look at your set up then it should make sense. If you need more guidance then just let me know. I have the orignial shop manuals for my 64.
