Thanks for the reply. I'm very new to this so probably have some dumb questions, and I know this is probably a rather silly issue since the motor runs fine, but it just indicates to me that there is a problem somewhere, but my worry is that (1) my alarms may not be working on the engine so I won't know if a bigger problem arises and (2) this is a potential indicator of a bigger problem that I really don't want to fully manifest 40+ miles offshore. I'm just trying to nail down all the possible failure points to diagnose since I won't be at the boat again until Saturday and will just have a day to work on it then until the next weekend.
Ground
I will check the ground. This is probably a really dumb question, but where is the ground connected on the engine?
Ignition Switch to Gauge Harness
The ignition switch is good and the connectors on the switch side are okay (since it worked with the other engine). So you are saying the connection between gauge harness that ties into the ignition switch and the gauges may be at fault? I think these are the wires off the clear plastic plug off of the ignition switch--not the three gray plugs. I can check those, as well as the connections to the gauges themselves.
Temperature Sensor
For the temperature sensor, I understand it is on the back of the motor directly in the block (between the heads). If the alarm goes off with the temperature sensor unplugged, is the problem with the temperature sensor? If the alarm does not go off with the temperature sensor unplugged, what does that tell me? I can always swap the temperature sensor from the good engine over to check.
Power to Engine When Key Is Not Engaged
I'm confused about the power to engine issue even when the key is not engaged -- the fact that the engine makes the faint humming sound regardless of key position (that the other engine does not make). And if I pull a fuse when the battery is on, but key is not engaged, I hear clicking noises in the engine.
Looking at other issues in the forum, any chance the main relay is causing the issue? Any harm or usefulness in pulling the relay over from the good motor to check.
Additional Info On the Problem
So to be clear, my ignition switch is the one pictured below. Previously, when turning the key to the on position (but before starting) the red and green lights circled would turn on, and I would get the two-beep alarm. These would stay on until I cranked the engine, then the green light would stay on when the engine was on. Now, no lights and no alarm comes on when I turn to the on position. But when I start the engine, the green light turns on and stays on. I understand that the circled red light is temperature and the circled green light is oil pressure, correct?