When the pump ran dry likely it lost its prime and just pumped air which it sounds like it’s still doing by the rapid operation and spitting. It can be harder than you would think to get a pump to reprime but once it is primed it’s fairly straightforward to get rid of the air and stop the spitting.
The good news is once the prime returns to the pump you’ll be fairly easily able to tell as the sound will slow and become deeper as the pump bites down on water.
The efforts for priming and purging go hand in hand, you may need to kinda push water into the pump any way you can. Make sure the water tank is fully full, this will create any head pressure that it can and connect a garden hose to the water system pressure inlet on the side of the boat and pressurize that.
You’re ensuring water on the inlet side of the pump and next dropping pressure on the outlet side of the pump.
Then while running the pump, open outlets anyway you like, try close to the pump and work your way away, open furthest outlets and work your way closer, try them one at a time, try them all open at the same time. It’s kind of a crap shoot here, you just keep pumping and flowing until you hear that change of tone in the pump that means you’ve succeeded!
More good news is these diaphragmatic pumps won't’ be damaged while awaiting water like the raw water pumps with their impellers. Hope this helps.
