You dont need a flushing T and you dont want public water pressure in you cooling system.
Remove both heater hoses at the engine ends and drape them over the right fender. Use a garden hose the flush first one way them the other, repeating until all the water runs clean and clear. Hook the hoses back up and try it.
Your car may have a water control valve in the right side that is vacuum operated. Those go bad and refuse to work even with sufficient vacuum , or you may see the valve arm move yet the gate inside it isn't moving. Hope this helps.
