Connect AC to the board and measure the amp draw. If none then connect directly to the heating coil. If still none replace the heating coil. I would not leave it connected since there is no way for the AC to be truned off. Also with the AC ground wire connected plug into a GFCI and make sure there is NO coil connection to ground. And a working coil does not mean the refer will cool as there are many other potential problems but it's a start. And turn on DC as needed for the board.
I've often recommended that RV owners buy, beg, borrow or steal (just kidding) a clampon ammeter. Be sure it will measure DC amps as the typical electrician clampon has no use for DC testing, but every RV or auto could. Also suggest 150A+ DC capability as some devices like a MW draw a lot of DC amps from an inverter. And inverter means exactly that 12VDC to 120VAC. The cost for a very usable clampon ammeter is well south of $100 and is pocket change compared to holding up a credit card. :B
