This is causing the relay to not pull in due to both wires being positive. But the other 25% of the time, the black wire becomes common ground and the relay to work correctly without touching anything. Could this be caused by a faulty circuit board?

What are wires tagged to on the other end, in the furnace/airhandler?
One should tie to common,the other to yellow post or yellow wire from the y post in the tstat

Something is wired in series with this circuit and is opening.
24V into contactor will read 24 V out of contactor until the output is allowed to go to ground. So a wire is loose or a pressure switch or other safety device is opening not allowing voltage back to neutral/ground.