Your description stating that removal
of the float switch wires causes the pump to stop after a few seconds is indicative that the control box and circuit are correct, and the opening of the contacts to the control box cause the correct response.
So you have proved to yourself, and as others have stated its the float switch contacts are sticking, however your own multimeter measurements of Open Circuit and Closed Circuit indicate that is not happening, which is where your confusion is why the circuit is behaving like it is.
You have not stated if the float switch and control box are closely connected, or via long run of cabling, but assuming it has no other suspect connectors in the circuit, then probably what is happening is while the multimeter measurements of Open Circuit and Closed Circuit as reported seem correct with float movement, with the small control current
supplied when the control box is connected, the float switch appears to be passing sufficient current
leakage (via salt
air junk around the float switch contacts) to allow the control box circuit to remain energised. You can prove this by putting your multimeter into milliamp reading and in series with the float switch and measure the milliamps as the float switch changes from open to close. If it's as described, you will not see much of change, and the pump will continue to run. OR the internal resistance of the meter will be sufficient that the control box will sense an open circuit and the pump will stop. As all the other posters have stated, marine
float switches are typically cheaply made, evil minded devices.
Additionally to fully validate the error is in the float switch: Connect a short piece of wire instead of the float switch on one connection at the control box, and then replicate opening and closing the float action with the short piece of wire to the other connection on the control box. We expect the pump to run when both wires end are connected, and then stop after the control box time delay when one wire end is disconnected. If neither of the above resultants happen then, post back.
Otherwise follow other poster's advice and remove and clean / replace the float switch. Follow a64pilot's sound advice and only buy good quality float switches and mount so they are readily serviceable.
The shower float switches are least of your worries, since the boat is new to you, you need to ensure your bilge
every time and you have a good spare on board for those.