Rotary tool: How to insert bits into collet straight?
it seems that you are inserting the bit in correctly, it just seems like it's not going in straight some of the times. Besides being careful when inserting the bit, I'm not sure how else to guarantee the bit is set straight without some additional tests. If you want to make sure the bit is in correctly without damaging your ornament, after you insert the bit give it a quick test on a scrap piece of material to ensure that the bit is straight. Or, if you can see that double-image effect, before engraving into your ornament just run the Dremmel tool and examine the rotation of the bit to see if you get the double-image, at which point reset the bit.