If mummies were the only ones to spread it, it would be an injury disease. All contact diseases can be spread by simply touching contaminated matter, which I interpret to mean creatures, possessions they were in contact with or their dusty remains.
Mummy rot is everything Leprosy was feared to be. I personally think the differences between the MM & DMG make a wonderful situation, where those who incur the true mummy rot also have their souls rotted away and are rejected by those who might otherwise help them[Cha] while those who are contaminated afterwards only suffer in body.
Yes, to me Slimy Doom = Ebola