This is just my 2 cents worth, and it is what I would do if this was my cutting. I know others have differing opinions and will come out with them. I grow a lot of these plants, I am not an expert, your mileage may vary with anyone's advice.
Yes I think your cuttings may be showing signs of rot.
Yes you did get cuttings that have the ability to root and grow...you got an internode, and an adventitious root (and a root nub).
I would take these out of water, treat the cut ends (ALL the cut ends) with a fungicide (I use Captan to pretreat all of my cuttings and rhizomes, before planting) and I would place these into very small pots, maybe 6", tied to a slim bamboo pole in a mix of very good potting soil (like Jungle Growth TM), perlite, peat moss, and orchid bark at a ratio of one part each, water them once, then let them lean against a wall next to a very bright light source with good air movement.
I would water once-trice a week because this soil mix is very fast draining. And I would not repot into a larger pot until I saw roots coming out the bottom holes. Then I would step up, using the same soil mix, slowly. Maybe to a one gallon.
Also be aware that some of your lower leaves may not survive, because these cuttings have to have time to establish before they start growing, and they may use energy from a couple of the leaves up as they do that.
