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.
If it doesn't have a control valve AND the hoses are quite a different temp then that just increases the chances of it being a plugged up core.
Seems to me anyway.
And heat going away at acceleration could just mean it goes away with more air flow over the core and getting heated less due to the increased volume. (maybe)
Another one of those 10 min to figure it out if we were actually there things. :D