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Hi my name is Robert....
IF you replaced the circuit board I would be looking at the limit switch, the limit switch could prevent it from starting. The limit switch should have continuity through it.
One other thing that catches my attention is the buzzing sound. Sometimes a power converter can put out AC ripple or dirty voltage and the circuit board does not like it, I would try to hook up a 12 volt source for the power supply and see if the buzzing goes away. IF so then the power converter could be causing this.
Click HERE for a wiring diagram you can see which wire is for the limit switch, the limit switch will sit inside the furnace near the combustion chamber, sometimes you have to remove the casing on the furnace to get at it.... IF you have more questions or concerns please feel free to contact me... Robert
You do want to make sure you have good DC voltage on battery power, check your grounds on the battery as well and the grounds where the circuit board is located. Has to be the limit switch, bad power supply or possibly the T-stat, normally you can jumper the blue wires on the back of the T-stat and the fan should kick on. My thinking is if it is running a cycle the limit switch might be preventing it from starting. I think the fan would come on so I would see if you can bypass the thermostat and see what happens IF it comes on and runs bad T-stat... Robert