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I have never seen a blocked heater core, but you can check it: with the engine normal temp open the hood and touch the rubber hoses on the passenger side, it's the warm water in and out of the heater, both hoses should be warm.
I suspect the temp control. More or less cold air is passed through the heater by a door in the air ducting. This door is controlled from the temp **** by a push-rod. You will find the door and the linkage under the dashboard behind the glove box (so best is to remove the box). May be the push-rod holder is not proper tight.