Recently purchased a "fixer upper" and the thermostat was dirty and falling off the wall. I decided to replace it with a Honeywell programmable. The old thermostat had wires for Rc, Rh, W, Y, and G. I put the new Honeywell on with exactly the same connections, and the thermostat reads about 10 degrees higher than actual temperature in the room.

I checked for dirt blocking the sensor, didn't find any, so I decided to replace the thermostat with a fancy NEST wifi thermostat. Contacted NEST to see if it would work with the Rc, Rh, W, Y, and G wires, and they said yes.
Put the NEST on exactly as they recommended, but the NEST is reading the same way, about 10 degrees too high. I removed the thermostat from the base to make sure I wired it correctly, and noticed the spot where the Y wire is connected is extremely warm.

I followed the thermostat wires down to where they connect to my furnace/AC, and saw that there are terminals marked Y, W, G, R, and C.

I noticed the Y and W wires from the thermostat are both connected to the Y terminal on the transformer. The W terminal on the transformer is not connected to anything.

The G wire on the thermostat is connected to the G terminal.

The Rc wire from the thermostat is connected to the C terminal. The Rh wire is connected to the R terminal.

Is having both Y and W connected to the Y terminal on the transformer be the source of my problem? Should I move the W wire over to the W terminal, or will that mess something up?