How about locating that 1/4" ply using small quarter round and basicly build a dado groove onto, instead of into, the side sheet. Could thinner ply be used on the sides this way?
PL or any polyurathane glue needs moisture to cure. When I've lived places where it's really dry, i've used a spray bottle to lightly moisten one or both surfaces. Have a care though as polyurathane glues will expand MORE with additional moisture during curing.