I like to believe that the reason the characters throw up ink is because they don't just melt on the outside. Ya know, for a little extra flavor :)
I imagine their insides becoming ink, but so slowly you don't even notice, but your body does, so as an obvious reaction to the unknown liquid inside your body that is toxic to you and could kill you, your body rejects it, and that's where the first caught happens.
The more the ink illness spreads in your soul, the more your body turns into ink, and ever so slowly you become nothing but ink, BUT imagine if some became lost ones. Or searches. Because like bendy ever so slowly appears to be more and more similar to his ink demon version in the game, I imagine some ink illness victims to become lost ones, but not like angelo who are ghosts and somewhat similar to lost ones
No no, like Alice in the game said, that she was a disgusting formless thing the first time, and the second time "it made her an angel", like the ink illness creates the lost ones. (Or searches.)