12 phones??? I thought we were talking about a big system.
Because every time you remove and then reterminate a cross connection, you introduce another possible failure. Because the wires get shorter. Because you might swap them by mistake. Because it's a waste of time.
Since the system is not working properly to begin with, what's the big deal in unplugging a few sets for a minute or two? You've already theorized that the current drain is causing the PS to fail. Now it's time to prove it by watching the supply with a meter.