I had the exact same problem a few months back -
I did the "hand test" and realized that indeed my heater core had become somewhat clogged - but not blocked (there is a difference).
Started off by buying two 5/8 inch hose splice kits each including two clamps and one brass insert each.
My remedy was to cut both hoses (1 one intake and 1 exhaust) from the heather core. I then proceeded to use a garden hose to flush out the heater core (you wouldn't believe the crud that came out of there). When I flushed, I did a multi directional thing, switching back and forth on both hoses.
I also use an air compressor to blow out more crud.
After the air, I poured some CLR into the heater core and let it soak for about 20 minutes. If you thought you flushed curd from the heater core withe the initial water and air - wait till you see what CLR does........ After the soak, I flushed, and flushed again withe wather and compressed air.
Total time counting soaking was about 40 min
Once I got nothing but good clean water going both directions, I spliced my hoses together. Brought the engine up to temp, checked for leaks -and now have had HOT air
(not warm - unless I want it) - double checked the fluid level and haven't looked back
NOTE -the CLR was used in the heater core ONLY
- and did not make contact with any other internal parts......