If both the hoses to the heater core are heating up, then the heater control valve is working and coolant is flowing properly thru the core. Sounds like the temp. blend door is NOT working and is stuck in the cooling position. That is the door/flapper you saw inside the housing when you were changing the core. When you change the controls from vent to floor to def, is the air blowing out the proper positions??
Say what??? Don't worry about the 'burping', unless of course, you're in the company of females...