So, now that I know the hardware is solid, I started in on the cabinetry. The cabinet is in three main pieces. The seat unit, the monitor housing, and the marquee cap. The Marquee cap, while having a LOT of angled rips and cuts, is the simplest and smallest of the parts, so I started there. I dismantled it slowly and took a ton of pics in order to help in putting it back together. The marquee itself is a large wrap-around thing with two heat bends that has a translite style art piece behind it. That is in great shape and, thankfully, will not need to be replaced. The rest of the marquee cap is ---steaming pile of meadow muffin--- and will need a rebuild.
As far as the art goes, my plan was to heat gun the heavy vinyl off the panels. That turns out to not be a thing. The vinyl heated through and wanted to stretch long before the adhesive softened up. I found a different way though - 91% isopropyl alcohol. It acts as a solvent to the adhesive without damaging the vinyl or the art. By soaking the edges of the particle board with it I was able to start peeling up the heavy vinyl and, once there was enough to grab, I was able to begin rocking the art piece side to side and eventually remove it. This does result in a lot of particle board residue still attached to the backside of the art though. This is fixed by liberally soaking the back of the art with alcohol and scraping with a plastic spatula. This is resulting in smooth, clean art that I can reglue to the new panels.
I'll be using the old, swollen, and separating particle board panels as templates to cut new plywood panels and then gluing the art to the new wood. This should result in a lighter and stronger assembly than original.