You eliminated the coolant as an issue since you have flushed it and refilled it and I am assuming you aren't having a leaking problem. The flush would run hotter than normal coolant because it's not designed as a coolant. It doesn't sound like your thermostat since your temp gauge and system readings are within range.
Sounds like a problem with your heater core being clogged/damaged. Both lines should be hot to the touch and since one is cold that indicates that fluid isn't flowing through it. Its purpose is to transfer heat into the cabin so that would explain your issue.