Heater problem: very little warm air in cold weather

Can you swap your heater hoses? Pit the "in" hose to the "out" side of the heater core at the firewall, and vice versa. That will send the pressure thru the core the other way 'round. In effect trying to flush out anything inside the core. Try it that way for a day or two to give it a chance to move anything out. If this gets you better heat then I'd drain the whole system and do a really good system flush. When you flush the system always swap the hoses to the heater so you can flush everything out from both directions.
Otherwise you are just "flushing" the crap harder into the core. Not actually "back"flushing it. Hope this works for you.👍