Go to HD or Lowes. Have them cut two equally sized doorskins or thin plywood, about 1/8" to 3/16" thick. Buy some 1 x 3 lumber, Pine or Aspen is OK. Screw the 1x3's onto one of the skins to make a kind of tray. For maximum strength pre-drill pilot holes. Put you piece in some plastic bags, then bubble wrap it. Make sure everything is as snug as a bug in the tray. Screw on the top. Make sure nothing is rattling around. Very high survival rate, this is how a lot of galleries do it for medium-expensive pieces. If you want, include a return label so the buyer can return the case. While it seems this produces a heavier package, you will still probably still be under the "dim weighted" equivalent weight that the shipper imposes on any large-but-light package, and you will pay the same.
I prefer to let the UPS Store handle my larger pieces. That way they're responsible for the hassles that follow damage in shipping. Which occurs around 20% of the time for large, cardboard wrapped pieces. I kid you not.
Also, for UPS and other shippers, there is a certain package size where you suddenly are hit with a pretty robust "oversize" charge. Know where that size is, which you can find out on their websites.