It sounds like you have an air pocket in the cooling system and the reason you have heat under acceleration is inertia is causing the coolant to flow to the heater under acceleration. If there is a bleed valve in the system you may be able to evacuate the air, otherwise try loosening a hose that is topmost in the engine compartment. If you get air out you will have found your problem. This often happens after a cooling system service or thermostadt change.
Good luck with it.