Since the pump pressurizes the entire hydraulic system, the grouphead solenoids are the only barrier to water existing the group. So if the group A drips, it's group A's solenoid, independent of why the pump is running.

Similarly your temperature idle problem could be the same root cause due to "thermosyphon stall". That's when water escapes by a slow drip after the pump has stopped, slowly introducing a steam bubble in the thermosyphon loop. That prevents the water from circulating between the boiler and group when idle, resulting in a cold grouphead until some water is reintroduced into the line.

Randy presents a thorough explanation in Understanding and Preventing Thermosyphon Stall. The solution, in your case, is either cleaning dirty/scale from the solenoids so they close properly, or replacing them if the coils/plungers aren't closing completely.