I transport paintings to shows and galleries proportionately more than I ship, but I have used a wood & laminate flooring underlayment foam (polyethylene) with great success. No dents, scratches or damage to any paintings since I began using it a year ago. It's very inexpensive, $25.00 at Home Depot for a 100 ft. roll. I simply cut to the size needed for each work. I cover all the painting surfaces (including corners) with the piece I've cut and sandwich the painting between two pieces of cardboard cut slightly larger than the overall size of the painting. Then I use the shipping style cling wrap is to hold everything together, it works well, nothing budges. As a shortcut, I've also used just one piece of cardboard for the back of the painting with the foam covering over the front, cling wrapped together and that has worked well also.
My oil paintings are always dry and have been varnished with Gamvar when I use this method.