Hi and thanks for using 2CarPros.com.
A plugged heater core will not make the engine overheat. Since you have a build up of debris in the cooling system, I suggest doing a flush again. Perhaps the debris was plugging the thermostat and not allowing coolant to flow properly. As far as a brand, discuss that with the people at the parts store. There are many to choose from, and from experience, they all do pretty much the same.
Here is a link that shows how to properly drain and flush the system. As far as replacing the thermostat, it needs to be there.
https://www.2carpros.com/articles/coolant-flush-and-refill-all-cars
Let me know if this helps or if you have other questions.
Take care,
Joe
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Monday, June 28th, 2021 AT 5:11 PM