First, running the system with 5.6V on logic is not good. It's above +10% already and a good PSU is one that keeps it voltages within +/- 5%.
Second, 11.4V is low but not dramatically so, things should still work. You could, as an easy thing to check, turn the 5V to some 5.2 and see what that does to 12V rail and does it help any. So middle-ground approach so to speak. But even if it does fix the problem, I'd say the PSU needs some work.

So as you've noticed old AT PSUs (and frankly also many cheap ATX ones) only regulate one rail, then scale the rest accordingly. Now typically you'd regulate the highest voltage but since in these PCs the 5V rail sees most power draw and is the most critical, it gets to be the regulated one. However, the scaling of the other lines should be almost linear, and in your case going from 5V to 5.6V should bring up 11.4V to well above 12V, and it doesn't.

Did you check the -12V (and -5 if present) rails, and not just on PSU itself but in the system as well? Too much load on these (a shorted capacitor) is not always detected by some PSUs and the power it can provide is typically low and won't blow stuff up, not right away. But the shorted line can affect the PSU outputs.
You most likely have dry or high-ESR caps on the secondary side and tons of noise on the rails, that can easily screw up not only regulation but also prevent things from working, and is dangerous in the long run (when the caps go completly the spikes can get really nasty). You don't need a pro service (though that would be nice, but expensive) to replace capacitors on 5V, 12V and -12V/-5V lines - just make sure you know how to safely handle a PSU like this (how to make sure the primary capacitors are discharged). To actually measure the level of AC on the power rails you need a decent meter, something with 30+ kHz bandwidth and preferably true-RMS.

It could be a drifted resistor in the feedback divider for 12V rail, but that should still provide a better linearity I think. That howerver is no longer a trivial replacement unless you know where and what too look for, measuring these resistors in-circuit will not be trivial, if at all possible. So if the simple recap job doesn't fix it, it's another PSU or a professional service. Which, again, is a good idea but not always cost effective.