If the heater core was bad, you wouldn't get heat at all. Since the engine is overheating, I have a feeling either there is air in the cooling system preventing the coolant from flowing or you have a faulty head gasket pumping air into the cooling system.
First, when the engine is cold, open the radiator cap and make sure the radiator is full. Oftentimes, people check the overflow reservoir only.
Next, if the radiator is full, again with the engine cold, leave the rad cap off and start the engine. See if the coolant bubbles or shoots out.
Take a look through this link as well. See if anything mirrors what you are seeing.
Let me know what you find or if you have other questions. Also, let me know how the cooling system was bled when you refilled it.
Tuesday, November 29th, 2022 AT 8:54 PM