CPU cooler appears to be corroded

First time posting, and hoping for some insight.

For starters, my PC is about 10 years old now. I'm very understanding that things could/will begin to go wrong for me regarding parts. A friend of mine helped build it for me, and for the most part it has run perfectly for me (hooray for me!).

The other day I moved the PC into the living room to play some games on the TV with the wife, booted it up and started getting everything ready to go. Though, after a bit the fans really started flaring up so I decided to just move the PC back to my office. I plugged it all back in and then boom. I get a message saying the CPU cooler fan wasn't running, and that the CPU was overheating. Big bummer for me.

I started up in bios, and sure enough the temperature on the CPU was around 89 c. I turned it off and let it cool down, and then started poking around my PCs innards for the issue. I thought at first it may have been the fact I had neglected to apply any new thermal paste since building the PC (though now I am seeing people say that's not all that necessary to do?), and went to Best Buy for some thermal paste. After cleaning the old paste off and applying new paste I booted the PC up. But alas, it was to no avail. It quickly began overheating again.

I've since resorted to just buying a new CPU cooler with the hopes that will change all of my problems. Though, looking at the old cooler, I have noticed along the edges of the copper plate there is corrosion (if it helps, the cooler is a liquid cooler from Corsair, I can try to find the model but I'm not positive I will be able to). I'm curious if there was something I did to cause this, or if it's merely the cause of parts being 10+ years old? I'm also worried there could be permanent damage to the CPU because of this. I'm willing to provide photos of the cooler for anyone curious that may have insight. Mostly, I'm asking if replacing the cooler is going to help, or if the PC is just toast?

Thanks for any help in advance!