When first started and placed into first, the car takes I'd guess about a minute to actually move forward. After that it shifts fine and runs great(now that I removed the double ground on the alternator).
Drop pan and clean or put in new filter. Look for lot's of gray gook or speckles of brass etc. If all look's fine Install new gasket and some new Type F fluid. See if that helps, if not got to go deeper inside for rebuild.
You are describing C4 "morning sickness" I bet it will go right into reverse when you first start it but it takes a while to move in the forward direction. If that's the case then you are very close to: 1. needing a rebuild. 2. ruining the forward clutch drum. It will take a rebuild to get it working properly. The seals in the forward clutch apply piston have worn or gotten hard over time and / or with heating. I would suggest that you pick the time (soon) for the rebuild rather than waiting for the complete failure of the transmission to decide for you.