The water pumps on these cars are pretty simple and reliable. It isn't a bad idea to replace the old water pump but it may not be your problem. I'd pull it off and inspect it. I suspect the t stat isn't opening. A few people have had bad t stats out of the box, you wouldn't be the first.

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2016 Corvette Stingray 2LT
1969 280SE
2023 Ram 1500
2007 Tiara 3200