You sure an empty outdrive reservoir won't shut down the engine to protect the outdrive?
Positive. Marine engines have to run. Even if that means running overheating with no oil in it and the sterndrive about to explode. They have to run. And the reason being is if you are off shore and you get caught up in bad weather. What is more important, loosing and engine, or dying in a storm because the engine won't start because some made in china sensor won't work correctly? That is the mind set with marine engines, so they are always supposed to, and designed to, run.
As to this. " When it's acting shorted, no electr0nics (radio, guages, lights, etc.) will work, and then a bit later they will work".
The gauges and radio are all separate of the engine. You can take the engine out of it and the rest of the accessories still work. But, what they do have in common is the battery. The engine does need to be connected to a battery to run, as the ignition system needs 12V always. If you take away the 12V, or if the voltage gets really low, the ignition coil will simply shut off. So if the 2 problems are to be related, than the issue is going to be at the battery. If you have loose connections, corroded connection, if the battery cables look ratty. Those will all do it. So have a look at those cables, if they are loose and corroded, clean them and make sure they are tight. If they look ratty, replace them.
Does all of that make sense?