Jord, this could be an LH module failing, but I seriously doubt it - and since you swapped it already, I think it's safe to rule that out. A sticking relay is also very unlikely, but easy to swap out (under rear passenger seat cushion).
However, it does sound a lot like what can happen when the 38-pin connector gets internal corrosion. Full details at this link:
https://www.500eboard.com/forums/showthread.php?t=10073
Check that X11/4 connector closely! If the pumps are running with the key off, try pulling the BM/GM and see if the pumps stop.
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