Heater blows cold at idle

I google this but only find stuff on 6.0. i live in south Texas so it's not terribly cold but my operating temp won't get normal until i drive for a while,and if I stop, it's starts cooling off. Leads me to believe thermo is stuck open maybe? Once it gets temp, which takes way longer than my 7.3 did, it finally gets hot, then if I stop itstarts to get cold again, if im idling and I rev it up, it goes back hot. Read it could be weak water pump? Has anyone had Same problems?

Can you monitor the ECT?

Engine coolant temp im assuming?? I have an xrt pro so no monitors if that's what your asking. Only have engine oil and trans on the dash thingy. this morning I started it remote and then went outside about 10ish minutes after and put key in and turned the heater temp to 90* and gave it some throttle and noticed it started to blow somewhat warm with even a cold temp gauge. Sign of water pump not pumping hard enough to get to core at idle?

Coolant level ok?

It has a leak I believe is on the inlet of the turbo line from the signs i see but I need to get a mirror and see for sure,but even when I top it off it does the same thing. At idle it goes cold, give it a little rpm and it starts going hot again. Doesn't seem like anyone has had this problem or has fixed something before they got this problem. I'm going to start with a new thermostat and fixing the leak and go from there.

Since there is a leak, you might be getting air pockets in your coolant system. That would cause what you are describing.