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Welcome.......great start. Continuity a simple and defining test. No continuity with the switch in the "on" position is defining. I have no knowledge of the circuitry of your boat from the factory. However.......bilge switches sometimes bypass the fuse block. My bilges are independent of the panel on my boat and have a dedicated circuit breaker from the battery. That way, with the battery in the off position in a multiple battery set up, or with the circuit breaker off (hopefully you have one before your fuse block) servicing the fuse block.....the bilges still work. Well worth your time to schematic all your wiring. Even if the switch swap works I say.......continue and explore. Is your switch a switch or circuit breaker switch????? If a circuit breaker switch what "cooked" the circuit breaker?