I have just replaced a power supply.
Did everything work before you replaced the power supply?
I wonder if there isn't a back emf diode in each striker solenoid and that they are reversed. An inductor can be expected to draw a heavy current for several hundred uS but not continually as Op's explanation would seem to indicate. OTOH a reversed diode would do just exactly that and with any appreciable length of wire attached could well draw the currents mentioned by the OP.Doc
Am I correct to assume that the high current that I am registering when energising this lock, is actually reverse current from the coil shutting down when it is de-energised, and this is in effect shorting the power supply and causing the reset ?
But before going further measure the winding resistance of the coil with a multimeter - if it really is 12V 0.35A, it should havea resistance of 34 ohms or so. Measure both directions - if there's a difference that implies a diode is present.