If the hoses going to the heater core are hot, and I assume they are considering all you’ve done, the problem is most likely with the HVAC system under the dashboard. The air temperature is controlled by a door, or flap, which moves (or should) according to where you set the temperature control. When you ask for warm air, the door directs air through the heater core. When you want cool air the air flows through the AC evaporator.
The device that moves the temperature door is often vacuum controlled. Engines at idle produce high vacuum. Engines at part or full throttle produce less vacuum, so if there is a leak in the system somewhere, the door can move on its own, depending on engine speed, causing exactly the symptoms you describe.
I’d start looking for a vacuum leak in the hoses or the vacuum devices under the dash. This is a very common problem. You may even be able to see or hear the temp door moving if you crawl down in the passenger footwell and do some experimenting with the throttle. NOT while you are driving, please. By the way, a service manual, such as Haynes, can be very helpful in diagnosing something like this.