heating system only gets warm when accelerating cool air comes out at idle.

Could be several things. Your blend door might not be opening except under acceleration because some are controlled by vacuum. It could also be your water pump's fins may be busted causing not enough coolant to be pumped back into the heater core. Under acceleration the water pump turns quicker. Also, less likely it may be a head gasket leak, pumping air into your cooling system causing an air lock going into the heater core. What I would do is idle the car and check to see if both the in and out heater hose are getting hot as well as both radiator hoses. Let me know what you find out from that.