You have fallen into a classic *XY problem* trap. You have a genuine problem:

> things are not working with the distance sensors.

That is not enough detail to help you with the actual problem, but I'll come back to that. You chose an (unfortunately inappropriate) test, and are now having problems interpreting the results. Unfortunately you don't realise that this is the wrong issue on which to be spending your time, as the result won't help you with the original problem:

> I'm using a multimeter to try and check what the problem is, and when I tried to check for shorts (with a mode called continuity, I think), one of the following may happen [...]

Unless you have evidence that there is a short-circuit (which often won't be the case, if you just guessing at what troubleshooting to do), then using a typical multimeter in that way on a board with ICs installed, will lead to lots of false positives *apparent* shorts. This is due to the ESD protection diodes (and other ESD protection structures) inside the ICs, which may conduct during that "continuity" testing.

So stop doing down this rabbit hole and back-up a few steps :-) Here is what I suggest:

 - Go back to your earlier configuration which you say worked i.e. external separate power supply for the 3.3V and 5V sections. I hope (and suspect) this will work again. This would confirm your hardware has no "short circuits".

 - Note that some other Sharp distance sensors have unusual power requirements. Specifically they pulse their IR LEDs at very high currents, for brief periods. Due to those high currents, they are well-known for needing lots of decoupling capacitance *close to the sensors*, especially if the power supply is weak, or has lots of inductance (e.g. long / thin wires etc.). Your sensor's datasheet mentions *average* supply current (12mA to 22mA) but not a maximum current!

 Therefore use an oscilloscope at the sensor's power pin to view the waveform, and you may find it drops below the minimum acceptable power voltage for the sensor (e.g. in brief dips, when the internal IR LED is pulsed) due to the factors I mentioned above. I expect decoupling capacitors will be required as a minimum. That may be part or all of the reason why "things are not working with the distance sensors" when they are powered from the DE0-Nano.

 - If you are still having problems with the sensors *only when they are powered from the DE0-Nano*, then focus on that and ask for more troubleshooting help. Show some photos of your setup, explain any constraints, use short power wires, expect to be asked to provide scope screen captures, and you will likely also need to provide info on the DE0-Nano and what power it can supply.