Here's a trick commonly used by specimen collectors and museum
Using thread or light fishing line, figure out a way to suspend the
piece of amber from the string/line. Sometimes, a fish hook with the
tip inside a bead hole is a good choice.
Suspend the piece from the top of any closable container, like a
shoe box. I like to use a small mason jar. I punch a small hole in
the flat lid, pass the thread through it and tie the thread to a
pencil. Let the amber hang in space. Then place a small container
(jar cap, etc.) inside the larger container. I just use the jar
itself, but pour some ethyl alcohol or isopropyl alcohol in the
smaller container or jar. Don't use denatured alcohol. Most rubbing
alcohol works fine. Close the container and place it in a warm room
The alcohol will dissolve the surface of the amber, fill small pits,
and leave a perfectly smooth surface that appears polished. When you
remove the piece, it's still tacky, so don't touch it. Just hang it
somewhere for a half-hour to an hour and it will be dry and shiny.