If your regulator is anything like the usual ones I see, I highly doubt there is a leak on the solenoid side, since that would be as you said "counter-intuitive." However, I would be interested in knowing what brand you are using to help further diagnose the problem. That being said, I would wager that your problem lies within the regulator, more specifically in the actual control knob. So instead of an actual leak, you probably have more of a "loose-knob syndrome" where, when backpressure is induced the valve rotates, too either more or less open.
However, just to make sure I would spray down your entire regulator with soapy water and check for air bubbles. This being said, DO NOT USE THE REGULATOR ON YOUR TANK UNTIL YOU ARE POSITIVE THE ISSUE IS FIXED, I have lost over $200 dollars of fish due to a faulty regulator.
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