Not likely that both sensors are bad. On this set up, you could start the engine out of the water without harming it, let it heat up while running slowly, see if the horn sounds once it heats up. As for that control box, it is one of the junk “fits all” models from OMC. I’m sure you could find the proper parts to repair it, or perhaps a used box. Fix this issue though, you sure don’t want the boat running away from you, perhaps hurting someone.
PS: please don’t let it get too hot, you surely don’t want to melt any of the plastic thermostat components or the rubber cooling water deflectors inside the block… Should only take a few minutes of running it at low RPM to heat it up sufficiently to set off the alarm…
