The spindle is suppose to be thinner then the spool core. It could be wobbly because the spindle is shorter then the spool or the thread is being jerked by the needle. This shouldn't cause any stitch problem though. The tension should take care of any slack. I use the Superior thread stand for all spools and cones. The stand will hold spools horizontal or vertical. It has never tipped over which makes it a great stand in my opinion.