If you have a traditional spinning hard drive (HDD) it might be contributing to the problem even if your games are not stored on it. Remove the power cord from the PC, open the side panel, and unplug both cables from the HDD (or all HDDs if you have more than one). Just check if that makes a difference.
While the Xeon X5670 is still a perfectly good cpu for general use on office productivity tasks, internet, video, etc. it has become somewhat out of date for gaming. Not totally outdated, however. Have you tried running your newer games at, say, Medium graphics settings rather than High? It's true that the graphics card does a lot of the heavy work when gaming, yet the cpu is also important.
The Xeon X5670 is a "Socket 1366" model which is, dare i say it, obsolete. There's no upgrade for your cpu. You will need, in effect, a whole new computer - motherboard, power supply, cpu, and RAM. If you build it yourself you could use the old case and the Vega card if you feel it's good enough.
One reliable measure of general performance is the PassMark Performance Test chart. For example, compare your Xeon X5670 results to a leading mid-range cpu of today:
Other things you can check:
What's using the most power? Open Task Manager and look at how much cpu % is being used when the computer is idling, as well as how much memory, disk, network, etc. Can you see anything odd or excessive there which needs investigating or disabling? Are there any processes or apps hogging your power? Also, look at the Performance tab. Here you can monitor how much "work" the cpu, SSD, and RAM are doing moment to moment. If you launch one of your games and choose the in-game setting for "Windowed" instead of "Full Screen" that will allow you to see the Task Manager window while you're gaming.
Another possibility, although remote, is that your old power supply is on its way out. Yet another is that the graphics card is failing. Any substitutions you can make will prove or disprove these possibilities. I would also shutdown, remove the power cord, and unplug then re-plug all the power supply cables that are connected to the motherboard (2 cables), the cpu, the SSDs or HDDs, graphics card, and the fans. Make sure all those are plugged in firmly.
If your system is as old my Socket 1366 / Intel Core i7-965 Extreme Edition it may be time to consider retiring it, no offense intended. Mine was still running OK when I gave it away last year, but I did replace the TX-750 power supply with a new Seasonic X850 model once over the years.