The plant in the HOB is called Lucky Bamboo. Not a true bamboo, but called so because of the way it looks. It's scientific name is Dracaena braunii (or alternatively called Dracaena sanderiana). Just keep the leaves above water and it will grow fine (it is a slower grower).
Usually when a plant has deteriorating/melt leaves, the damage on the leaf is unrepairable and is best to cut those damaged leaves off so the plant doesn't waste energy trying to repair what can't be repaired, rather than use/focus that energy towards new growth. I'm not sure if the melting leaves from emersed plants transitioning to submersed growth also wastes energy trying to repair melting leaves or not. Just to be more certain, I would just go ahead and cut off deteriorating leaves (they are gonna melt/die anyway).