The clutches that Club Car used from 1984-1987.5 were terrible. They came out with a much better design halfway through the 1987 model year. If I were you I'd upgrade both if they're the original clutches. The original 1984-1987.5 drive clutch did not have a bearing in-between the two sheaves for the belt to ride on, which is likely contributing to your cart's wanting to start moving before the engine fires up.
The other thing you could do is make sure that you have the correct drive belt. The older clutches used a narrower belt than the newer ones, so if someone put a newer, wider belt on that older clutch...that could also cause it to try and move as soon as the starter/gen begins turning (as the wide belt is already in contact with the sheaves on the drive clutch prematurely).
One thing to note if you decide to go the replacement route; the drive belt center lines between the two clutch styles are different, this is why I say that you should upgrade both.
