On your era furnace, all of the power to the circuit board goes through the thermostat, limit switch, sail switch as well as the thermostat wiring. Each of these items can easily have a voltage drop acros them.
Easiest is to test, clean or replace the thermostat. You can test the thermostat by turning it on and measuring the voltage between in and out- you should get zero, but I would bet you will find a good half volt, at least.
FWIW, the furnace should run down to around 11 .5 volts, though 12 should be the limit.
