The G wire is not so G!
The G wire setup
The G signal controls the fan. Your thermostat is capable of running the fan on its own without heating and cooling applied for the circulation. As long as you have this signal present and connected to the thermostat of course!
Traditionally the G signal is considered a cooling side signal. You might already see some problems here…
Problem #1
The G signal will be powered by the cooling transformer and not heating transformer as intended.
Possible Solutions
Use single transformer setup
Check your thermostat manual to see if it is possible to reassign this signal to be controlled via heating transformer
Do not connect G signal. At the price of losing ability to run fan on its own
Problem #2
The G signal will be enabled whenever you turn on the evaporative cooler to cool your house.
Possible Solutions
Use single transformer setup and leave the G signal connected if you are okay with having extra circulation during cooling
Do not connect the G signal. At the price of losing ability to run fan on its own.
Move G signal to W1 connection and W signal to W2 connection. Might be not possible if both of those signals occupied already!
The ideal and possibly not realistic solution
Keep dual transformer setup
Set your transformer to control G via heating transformer. (Might be not possible)
Turn off furnace power in the summer to disable extra circulation (if desired)
Turn off swamp cooler power in the winter as a safety precaution
