If you think the core might be it, you can try back flushing it. They do sometimes stop up. But that does not explain why the heat goes away on acceleration. Back flushing it would be a LOT easier than replacing it since the dash has to come out.
Luckily, that's something I don't have to worry about. On my model of Ranger (1992), the heater core is about a 30 minute job. The dash doesn't have to come out. It's accessible behind a trim panel under the glove compartment.
I am not discounting anyone's suggestions. I will take a look tomorrow when I get a chance. Thank you to all for your suggestions, and I'll update as soon as I have more information.