Transmission does not engage first or reverse on start, will act as in neutral?

Won’t engage first or reverse on start, will act as in neutral until it’s almost at operating temperature. Have to feather throttle until it smooths out then it shifts fine. Hard stops make RPMs hang at 1,000-1200, will rev then jolt into first if I try to accelerate before it stops hanging. Haven’t changed transmission fluid ever. When accelerating after getting into second it will not go into third properly, can get it to 4,000 RPMs if I wanted to but if feathered around 2,500 it will shift poorly eventually. I don’t know if this is a TPS issue or what, also sounds like someone threw rocks in the transmission at idle but not positive on the location of the noise yet. Any insight would be appreciated before I take it to the Toyota dealer. EDIT: After driving for a while, it’ll shift great except for hard stops.