The pump operates continuously when switched on. The switch on our 2005 boat just below the sink in the head. There is no float switch that activates it. Any small bilge pump should work if it fits. Try Catalina Direct.

For future reference, just replaced mine yesterday. Used a Rule 12V 500GPH pump (without float switch) and crimped to the flying leads from the pump switch. Screwed the basked to the shower floor and hose-clamped the output hose on. The whole job took less than 10 mins. 

On mine the problem wasn't the pump, it was the switch.  Once I rebuilt the switch it worked fine. 

I installed a Rule auto float switch model, low profile that is wired to its own panel switch and is plumbed to the head sink drain thru-hull pipe in the loop that snakes up above the water line behind the toilet.  Two one way valves in-line down in the sump, one before a twist-off covered strainer and one after. Works great although you do have to monitor the strainer as any hair that gets into the one way valves can leave them leaking. 

I have a question about the function of the pump.  The boat is new to us and I ran the sump pump last week to check the function and seek out an odor source.  When I run the pump, it first backs up into the sink before it drains.  The shower drain and sink drain share a thru hull.  Is that normal for the sink to back up like that?  Thanks.