Round honeywell thermostat

The hermostat is a switch contact that works n the selected temperature, nothing more elaborate.
For a wireless thermostat, all you need is power fr the receiver and the cntact connection. The thermostat transmitter works on battery power.

Those normally do not fail, they have a mercury switch in them. However there is a cravat they have to be mounted level and up side up. If that is the way it is mounted then just replace it.

If you are talking about the difference between heating mode and cooling mode and the room setpoint, you can not change it. It comes from the factory with a 3 degree difference. In other words if you raise the heat setpoint until it is within 3 degrees of the cooling setpoint, for every degree you raise the heat setpoint the cooling will raise a degree.
About all you can do is increase the distance between the heat and cooling setpoint. To do this either lower the cooling setpoint to 3 degrees higher than the heat setpoint you want and then increase the cooling setpoint the desired setting. For example heat and cool setpoints are 67 and 70 respectively, and you want 65 and 73. Lower the cooling setpoint to 68 and then raise the cooling setpoint to 73. Now your heating will come on 3 degress lower than 68, the 65 you wanted and cooling will come on at 73. Remember, the thermostat new setpoints will turn on the cooling at 75 and turn back off at 73 and heat will turn on at 63 and turn off at 65.
You can perform the same using the heating setpoint, just have to look at it backwards. If you need to lower the difference I recommend a 8000 series thermostat. They are the most functional thermostat that Honeywell makes, be in mind they are about 3 times more expensive and many of the setting require a well trained tech to interpret and set properly.
Hope this helps.

the burners light? if the burners dont light than it's working right the blower on the furnace comes on with the a\c

heat anticipator is shorted or broken. if this is the case you must replace the tstat.

the power for that stat comes from the furnace - if you have no display on the stat you have a furnace problem

If you turned the ring and the pointer moved, then you are using the thermostat correctly. Turn the thermostat up above room temperature (the temp on the bottom of the thermostat) Make sure all your switches are on, and that the fuse or circuit breaker is also on. If you still do not have heat after verifing all these things, then it is most likely a problem with the heating appliance.
Also, if it is a heating and cooling thermostat, make sure your lever is in the heating mode. Hope this helps

The thermostat will not cut off the A/C until the thermometer reading gets to 76.

If there is not a heat source near/behind the thermostat which is causing the thermometer to read higher than actual room temperature, it sounds like the thermostat thermometer is out of calibration.

There are ways to calibrate it, but much to involved to go into here.

One solution is to just set the thermostat at 80, knowing that it's really set on 76.

Another solution is to replace the thermostat.

Charlie

try to get one that is made for oil heat, the installation is easy only two wires, and the instructions make it easy.

The honeywell tradeline thermostat would work as a direct replacement (no mercury in the new ones), however if your using just heaters and no cooling system off of it, you could Heat only thermostat from honeywell that would replace your unit as well. Both at available at Ace Hardware with the heat only unity being roughly 7 dollars cheaper.