In a situation like this building a custom sensing board might be a little excessive. If you wanted you could hack into a commercial electronic backgammon board like this one. I have no idea how it works, although I’m guessing the board has pressure-sensitive switches you push down with the game pieces as you move, but you’d have to figure that part out yourself. I did something similar with an electronic chess board, but it took me many hours with an oscilloscope to figure out how it worked. Maybe just playing against the built-in computer would be the way to go for now!
-Adam