It sounds like you've not got the worm gear engaged properly. There worm gear is on an eccentric, so you can rotate the motor mount etc to push the gear into a tighter mesh, which will substantially reduce the backlash. There's a screw you undo, then the motor will rotate freely... it probably says in the instructions.

The rotary table you linked to wont be any better than the one you've got if it's adjusted properly.

P.S. You might want to put a grommet round the motor cables so that the aluminium plate doesn't cut them.