Having re-read this several times let me ask you a daft question.  You are using the screw part in the correct fashion ?   The line about the screw getting stuck made me think.

The threaded end is for screwing into the top part of the desk and you would lower the desk lid and the screwed in bits into onto the base. The screws will align with holes and will need wiggled to slot into them. The cam is then inserted into the holes in the base and turned to grab and pull the screw head in, therefore making a solid join between the wood.   If you are simply sticking the screw head end into the holes on the desk lid and then trying to grip the threaded end in the cam lock then that's the issue.  The cam lock grips the top of the screw - the end that you use the screwdriver on to drive it into the wood. The threads are for screwing directly into the lid.

If that's not the case then post pictures please as the descriptions mean not a lot