Sounds like irreversible damage if the surface is 'melted'. Sorry to hear this.
Is the white merely smeared white lithograph ink (try a Mr Clean magic Eraser on it) or is the surface of the plastic visibly melted and marred?
This is how we all learned our lessons. Different plastics & recognition of their properties is one of them.
Citrus based gum removers like Goof-Off and Goo-Gone (environmentally and PC friendly as they may be) are highly corrosive to many shiny hardshell plastics used for contemporary vacuum cleaners and are just about the worst fluid on styrene-based plastics because they eat away and dissolve the plastic surface. Depending on the plastic's compositional blend it is not always evident at first swipe. Vinyl-based plastics such as polypropylene are virtually impervious to citrus cleaners but ABS is usually not. Both types of plastic are used on the DAMS such as the back but the bag cover is styrene-based, iirc.
Grease pencil china marker usually comes off cleanly with soap & water.
To remove 'permanent' Magic Marker or Sharpie ALWAYS first try a vigorous scrub with a Mr Clean Magic Eraser. This removes most of the surface ink and depending on how long the marker has been on the plastic, sometimes the marker has reacted and left a faint stain.
If the mark still faintly shows follow up with a fresh Magic Eraser.
Only then do you get out the adhesive remover.
The ONLY gum adhesive remover I will use on a vacuum is Naphtha aka Lighter Fluid such as Ronsonol. It will NOT attack the plastic but may soften the printing so careful application is essential around silk screened logos & such.
All the alleged gum adhesive cleaners on the market have warning labels on the back; it is always best to err on the side of caution and always always always do a test patch on a hidden inside surface when in doubt - which should be EVERY time.
Ruined and defaced the front of my Kenmore Portable AC unit cabinet when a friend decided to help out removing the peel & stick marketing label panel on the front with Goo-Gone; the surface now looks like a burn scar.
Just as Varsol and Turpentine have different properties for cleaning oil paint brushes, so too are Naphtha and Citrus Cleaners entirely different solvents.
Similarly, many people believe the advertising that WD-40 is an all purpose lubricant, rust-inhibitor and price label sticker cleaner. In fact it has exactly the opposite effect if left in place for any length of time.
WWD-40 is permanently banned from all my work stations.
I'm sure others here can post their fool-proof favorites that never fail.
Dave