is this 12V power supply functioning properly and will it power my device?
It sounds like your power adptor is OK.  There should be no connection between the AC input pins and the DC output.  Whether the two DC output terminals should show continuity or not will depend on the circuit inside the power adaptor.  I wouldn't be surprised to measure continuity with one polarity of the meter, and not continuity with the other. (Measuring the continuity/resistance of the output of a power supply is fairly meaningless - measuring voltage is much more useful.)

The continuity test on a multimeter indicates a low resistance connection (perhaps under 100 Ohms) between the probes - not necessarily a short circuit (zero Ohms)

Your suggested connections to the DE-9 connector are correct, assuming that the red wire from the supply is positive.