Thank you for the compliments.
Anything carved is going to be cut into the glass and you get to feel that depth. The only way you would be able to make it a flat surface again would be to infill it with something (wax, epoxy, etc.) Filling it would cause it to go back toward the clear side (or a color if you used one) and it won't have that stronger contrast for viewing. The appeal to the carving though is by creating the refractions of light through the etching which highlights that depth.
It's good to have that ability to feel the etching/carving. Having the piece in hand allows someone to appreciate the artwork more by touching it adding more use of their senses.
When I etch on both sides of a coaster, I generally use a surface etch which keeps it smoother on top. I do carve deeper on top in certain designs and usually nothing that would interrupt the drinking glass if it is used. The coloring will be different when you aren't looking through the glass. That is an extra way to use shades as well.
Best of luck with your carving!