Thoughts from a C-320 owner...
Michael, this is a great group and helpful. There is an additional great resource for C-320 specific issues that can be had by joining the Catalina 320 International Association also.
The tank valves are not a likely source of the problem on the C-320 because the forward tank is higher than the aft tank and leaving them open simply means that the water drains from the forward tank back to the aft tank by gravity. When you're out of water, you're out and absent a leak in the system you won't suck air until then.
The most likely source of pump cycling is a small leak downstream of the pump in a line connection or a valve. It doesn't take much of one to keep the pump cycling. On our 320, it has always been the cockpit shower valves. If a valve is left the slightest bit open, the pump will cycle and there will be little observable leaking until you know what to look for.
Our washdown pump will develop pin hole leaks in the hose over the winter (maybe someday I'll learn to remove it) and they are hard to see. If the pump is cycling, I always know I have to trace it down.
If it isn't a leak downstream, it could be a leak upstream sucking air, but that usually causes a loss of prime and reduced or non-existant flow and is easy to diagnose but harder to find.
The fittings can come loose in the system up or downstream and there are clear instructions on the Association website for re-connecting both types old (yours) and new. I don't recall any check valves in the system, but there is a trash filter at the pump. Again this usually causes flow problems not cycling ones. Pressure switch failure tends to be all or nothing.
Lastly, while in the neighborhood, the inspection ports on the water tanks tend to leak when full. The front tank's water will go to the bilge if both valves are open and the aft port is leaking. Teflon tape, doubling the gasket and a certain plumbers putty recommended on the Association site archives solve the issue.
