Straight answer:
The only way to measure the true current flowing on the 12V rail is using a Clamp amperemeter, also called a Current clamp. To be specific, you need one that is able to measure DC currents, so one using Hall effect feedback loop, preferably with a digital display.
See https://en.wikipedia.org/wiki/Current_clamp
These are not cheap, but still affordable.
To measure the current, you select all the yellow wires from your ATX power connector or a PCIe 6 or 8 pin connector or a HDD power connector and clamp the clamp around them and around them only. These yellow wires are used for 12V. Similarly, red is 5V and orange is 3.3V. Black is GND.
Then you see the current on the display of the current clamp.
Done.
Further thoughts:
If you can't get a hold of a clamp amperemeter, then using a cheap power meter between the wall socket and the power cable will give you the total power used by the computer. The PSU efficiency ranges from between 70% to 95% based on the 80Plus rating of the PSU. In addition if you assume that 90% of that power goes over 12V rails, you'll be right for a modern system. So
wall_socket_power × psu_efficiency × 90% / 12V
is also a good guess for the currents going through the 12V rail.
Nowadays, most of CPU, memory and mainboard power comes from the 12V rail, as well as hard drive and external PCIe GPU power. Non-GPU PCIe cards typically use the 3.3V rail. The 5V rail is largely unused, except for HDD electronics in some cases. Your calculation is right for the peak case in this regard, including power conversion efficiency. 90% is optimistic, but possible. You should add 3W for each stick of memory and 5-10W for mainboard chipset, though.
Altogether, your PSU appears to be enough for normal usage and the only moment when it'll get anywhere near its full capacity is upon boot, when the drives are all spinning up at once and the CPU is not yet programmed for power-saving and runs full-throttle. If this becomes an issue, the drives can be configured to be spun up in a staggered manner by the OS.
If you're suspecting the PSU to be a reason of unstability, attaching an oscilloscope to the individual voltage rails and watching for voltage fluctuation under different loads is a much better indication of an insufficient or failing PSU than measuring the rail current.
Enter your firmware setup, set it to Advanced Mode if you haven’t done so already.
The setting is located at: Advanced → CPU Configuration → SVM
Upon saving, the PC may turn off completely and turn back on again, if I remember correctly. This is normal.
This information is, naturally, available in your board’s manual, page 3-15. ;)
Best Answer
So right after I posted this I went to the same page and loaded the "Optimized Defaults". Several options appeared in the advanced menu, including SVM, which is enabled by default. I don't recall making any custom settings in the BIOS previously, but this seems to have solved the problem. Thanks!