Dell Optiplex 3010 startup issues, random shutdowns, no diagnostic lights

Hello all. This has been bugging me for a few days, and I can't find anyone online with this exact problem.

My Optiplex is suddenly turning off after random amounts of time running fine, and sometimes has trouble starting up. For instance, one morning I had to press the power button multiple times to get the computer started (many instances the computer starts for maybe 3 seconds and shuts back down). Once it was started, I was on for roughly 30 minutes before it shut down. It didn't BSOD, it didn't really seem to "crash," but it acted as if I had manually went and pressed the "Shut down" button, giving me the full "Windows still has programs it needs to close" popup (or whatever that shutdown popup says). I turned it back on and this time it ran just fine for hours before, randomly, shutting off while it was sleeping. I can't get a consistent read on the diagnostic lights -- sometimes they all flash before the computer starts up, but I doubt that means anything. I'm not running intensive programs or anything like that and I've never had any issues with overheating. I'm using 2x8 DDR3 RAM, which I know is above the recommended amount for this machine, but it has been working fine with it for well over a month. In addition, the issues are not while the computer is running. When it's on, it runs like it always has. The only issues are sometimes it has trouble turning on and sometimes -- very randomly -- it shuts down.

With these issues I'm really not sure what to call it. People I've talked to say it could be RAM, could be hard drive, could be motherboard -- but from what I've read online and what I've discussed with others, a PSU issue seems to fit the bill better than any other.

That leads me to my questions:

Does this seem likely to be a PSU issue? I am using a Corsair CX430 for power, and have been for around 5 years.

How can I know whether or not I'm just missing the diagnostic lights? I get the feeling that I might just not be paying attention for the right thing.

Are there any other tests I could do?

Any help would be appreciated.

Typed on my Optiplex 3010, which for the moment is running just like it always has.