Your power supply may be one that requires a minimum load. OEM supplies intended to be embedded into equipment where the load is somewhat known would often be like this. Typically the minimum load is 10% of the maximum load, but of course only the datasheet can tell you for sure.
Since you don't seem to know anything about this power supply, try putting a 5 Ω or so power resistor across the 5 V output and see what happens. If that seems to help but still doesn't quite get you there, try 2 Ω or even 1 Ω resistors. Be mindful of the power these resistors need to dissipate. The 5 Ω will dissipate 5 W, the 2 Ω 12.5 W, and the 1 Ω 25 W if the supply output is actually held at 5 V.
In 5 V and 12 V supplies, especially those intended for computer use, the 5 V output was often considered as the "main" one. It was tightly regulated, as long as the load conditions were met. The 12 V output then is what it is, but usually not more than 500 mV or so off from nominal. You probably don't need a minimum load on the 12 V, but you'll just have to experiment unless you can find the datasheet.