Your multimeter showing continuity doesn't necessarily mean there is a short.
All your multimeter is doing is applying a small voltage and then, if the current that flows is over a certain threshold, reporting that there is continuity.
The components (stepper drivers, microprocessor) on the board, draw current. That's normal. The current draw might be enough for your multimeter to report continuity. Because the multimeter is only using a small voltage to test (and not the required 12V), the current draw may be intermittent (not enough voltage for the microprocessor to actually start working), and capacitors getting (dis-)charged may also affect things.
The multimeter not reporting continuity is a guarantee of no short. However, the multimeter reporting continuity does not mean there is a short. The only way to find out is to apply 12V. If you use a current-limited power supply the possibility of damage if there is a short circuit is limited.