Red wire from connector is for the gas valve. That sounds correct.
Blue wire is receiving command from thermostat. That is correct.
White wire is receiving command from thermostat. That is correct.
I would still clean the elctrode/ flame sensor. Inspect high voltage wire. Have helper turn on furnace as you listen carefully to see if clicking is in ignitor area or in high voltage lead indicating bad lead.
Do you hear any burner coming on?
Could be dirty orifice in gas piping?
Look at your water heater circuit board and compare it to the furnace circuit board. Some are the same. You could switch it out to test to see if that corrects problem.
Does your furnace have the exterior door for access?