Many bottle stoppers that are carved from wood are for wine bottles. These stoppers are composed of the carved wood top, a dowel, and a cork. The dowel is 3/8 inch and the wood top and the cork have a 3/8 inch hole drilled in them for assembly. I have found that the corks sold by Woodcraft stores do not really fit tight into the bottle top. Here's a solution to this problem.
Obtain the synthetic "corks" that are used in wine bottles. I asked friends to save them for me. These "corks" are longer, and wee bit wider for a much better fit into the wine bottle. Drill a 3/8 hole all the way through. Add a 3/8 dowel and use your rotary tool with a used sanding sleeve to increase the "corks" taper.
In addition to "cutting" a taper on the synthetic cork you may find that the process has rejuvenated the sanding sleeve by removing embedded particles.
Assemble and glue up the three pieces (carved top, dowel, "cork") . If you need to you can cut the bottom of the "cork" off if it's too long.
This method will produce a tight fitting bottle topper.