Shoe,
I made a similar cheesepressing frame, using hardwood for the top and bottom, and dowels to keep the wood level and centered and had the same issue with the top board binding.
Rather than drill the holes to be larger, I used my Dremel with a grinding bit on it to sand the holes just large enough that the wood moves up and down freely, without binding. It didn't make the holes large enough that the top board would slip, slide, or tilt, just enough for free movement.
Good luck and an early Happy Birthday!