EDIT: (after reading your last post) Yes you can certainly try jumping out the float switch! You should wire in your new thermostat first though (or wire your old one back in if you intend on taking the new one back if the old one is not bad).
Ok this is a bit confusing then. Those jumper settings indicate your old thermostat was set to control an electric heating air handler where the electric heater
brought on the fan in heating.
Because the middle jumper was on Gas/Elec and NOT HP, there would be no control of anything attached to the O/B terminal where your blue wire was connected.
The way the jumpers and wiring was on your old thermostat seems to indicate that you have a traditional cooling only outdoor unit with a traditional electric heating air handler. The way the old thermostat was wired, it would work fine this way. The only oddity about it is the blue wire connected to O/B on your old stat would not have done anything.
With the info I have now, and without actually seeing pictures of how the thermostat wires are connected at the air handler, I have to instruct you to wire your system as a standard electric heat with cooling system.
So on your RTH111 you would wire Green to G, Yellow to Y, White to W/Aux and Red to both R and Rc (wire to R and leave the factory jumper to Rc). You can tape up the end of the blue wire and tuck it back into the wall, you will not need it.
If your system doesn't work when wired this way then you may have originally had another problem it. I can try to help you diagnose that but it may require you using a volt meter and taking voltage readings with the system powered up. Let me know if you're comfortable doing that.