if its just a move and not shipping is involved the wet paper and plants isn't necessary. We ship plants like that because water weighs a lot and minimizing the amount makes it cheaper to ship. When I bring plants to a convention or deliver them in person I just put plenty of water in the bag and then knot it far enough down that the plants are all kept wet as the water sloshes about.
I have moved tanks both ways, but not huge distances. Of course its a lot easier moving a tank with gravel and plants when its a smaller size tank. As noted there is a risk, so you have to decide what you want to do. Safest is to empty the tank but it is a lot more work.
Bear in mind that tanks need to be supported completely around the entire bottom rim. They usually need to be pretty level as well. Lack of support or not being level will both cause tanks to crack or shatter. Basically not supporting or being tilted causes the pressures on the glass to change from an even distribution to not so, and when the glass can't take the pressure, it cracks or shatters.
What is even more freaky is that glass can actually bend. The first time I looked across the front glass of my 45 gal tank from the end and saw it was bowed, I almost freaked. But a bit of research showed me glass does have a little give and this bowing is normal.