It’s not an easy one to help with from here.
Basically the PSU is the most important quality part in the machine, but also one many manufacturers skimp on and find a huge variation in quality between brands with not a lot of external signs for the nongeeky.
You could think of it as analogous to tyres on a prebuilt bike.
At a guess the old one died from being mediocre and old, and the new one died from being crap (or VERY misleadingly output rated(common indeed)).
Most PSUs have output protection, even the crap ones, so I would imagine it is not the rest of the machine or it would never have worked.
If I were you I’d take the old one back and get a refund, then buy a quality brand.
Do you have the original one so you can inspect the output power for each rail? This is usually the cause of the problem, insufficient current.
Brands I would recommend are Seasonic and Corsair, but as these are real quality they might be quite expensive…. But then they tend not to drop dead or destroy the PC when they do.