Ouch... how many times have motored back with no water, the damage could already be done...
With that said, you have to remember that your engine cooling system is actually TWO systems, one "Open Loop" and one "Closed Loop"
The closed loop is the part of the system that circulates anti-freeze/coolant throughout the engine to keep things at a mean operating temperature. It''s has an impeller and thermostat, the impeller for circulating the coolant, the thermostat for controlling coolant flow in regards to engine temperature. It is very much like an automobile system. Just remember to change your coolant about every other year. You can check your thermostat''s operation by putting it in a pot of water on your stove, with a suitable thermometer. Heat the water, and as the thermometer reaches the thermostats operating temperature, you should see it begin to open, and then open fully above the target temp. The closed loop part of your system will have no effect on the pumping of water you see coming out of the exhaust.
The open loop part of your system sounds like the trouble to me. It has a few more areas that can be troublesome. It''s components consist of: Intake thru-hull, seacock, strainer, impeller, manifold, muffler, exhaust pipe and connecting hoses. Each of these alone, or in combination could cause your problems. The open loop part of the system is what keeps your coolant in the closed loop part, well, cool. It takes the place of the radiator in your car. The radiator acts as an air-to-water intercooler, your manifold acts as a water-to-water intercooler. If your open loop is not pumping water, you can overheat your engine and damage will result.
Check each part of your open loop CAREFULLY. Is the thru-hull blocked? Strainer full? Any cracks in the hoses? Is the raw water impeller good? (remember you have TWO... one for closed loop, one for open loop) Is your muffler collapsed or damaged? Basically there are two major areas to check, one is on the intake side, where a leak can cause air to get into the system and cause the raw water impeller to cavitate. On the exhaust side, a blocked muffler can cause the raw water side to stall.
I always check my system over thoughly every spring and replace any and all suspect parts.
Hope this helps some.