. At this point I'm looking at dismantling the ice box cabinet to replace the it and hoping someone has insight on how to do this without destroying the wood.
If you are convinced that the problem is inside the top loader ice box cabinet, I would cut large round holes on the aft surface of the cabinet (next to oven/cook top) as well as the surface under the galley sink to inspect & clean before taking the entire cabinet apart. You may discover that the problem is not inside the cabinet. You can then fit plastic port hole type inspection plates that will provide access in the future and will not affect the visual appearance of the cabinet. The cabinet is likely glued & screwed together & not likely to be disassembled without major damage To the wood.
I agree with
@jssailem that the biggest problem is the seal.
Do you use the top loader continuously? If not, turning the ice box off & leaving the door in upright position, when not aboard, to promote air circulation as well as wiping seal with mold retarding cleaner might be a simple solution. If you tear apart the cabinet, I believe that you will regret the outcome & not really fix the mold problem.
I installed vents (purchased from Beneteau) on my refrigerator cabinet along with a fan behind the vent to exhaust heat from the compressor located inside the cabinet. Perhaps ventilation would reduce mold growth, if you do find mold inside the cabinet.