For detecting where short circuits actually is, you can theoretically use DMM, but most of the DMMs have problems with resistances below 1Ohm (mostly limited by simple two-wire probes) and do have only optical indication of the value (=you must jump your eyes between PCB and DMM), so it is not that great tool for exact localization of the short circuit.

Having tool to distinguish between resistances in order of tens miliohms could enable me to find where the short circuit is. The tool doesn't have to have ability to display the exact value, only indication if the resistance goes down/up as I move electrodes closer to/further from the short circuit location. Sound indicator (buzzer) would be great for this, as I can fully concentrate my eyes on PCB, while beeping (low tone=low resistance; high tone=high resistance) would indicate the resistance.

My first idea is something like this: three resistors (R1 mostly adjusts the measurement current, R2 limits the voltage on measured resistance; R3 produces slight voltage offset to allow differential amplifier to work correctly), differential amp itself (four resistors and opamp), trinket pro, buzzer, pushbutton (to allow "zero-ing" the "measurement") and optionally display.

By the way, I feel like four wire method will not be actaully needed, especially when probes will not be detachable, but I have to make some experiments with this.