I had a couple of rooted cuttings that did something similar (lost their vitality, stopped growth and the green bits were yellowing/blackening, though I don't recall any softening). I carefully extracted them and found necrotic tissue that had once been roots on some of them. Two had fungus gnat larvae that had eaten the young roots. Several others were sent to me with roots already dying. One had just one tiny root about 1/4" long. In all cases (except for the one with 1/4" root), I let them dry and then basically started over with an attempt to optimize rooting conditions. I used almost pure perlite as a rooting medium, in cups with LOTS of ventilation holes in the sides and bottom, with just a little bit of soil added near the top to provide small bits of nutrient in case any new roots formed in the "restart" attempt. Then I paid attention to water, heat, and light.
The two with fungus gnat damage never made it (they were goners by the time I noticed what had gone wrong). Of the others, surprisingly enough one of them made it. It was probably just because I caught it early enough that it still had energy to "restart". (Also the one with the 1/4" root made it... that was a different problem I think... had been given to me about 3 months after starting, and it was alive but "stalled" and not thriving... it too responded to this treatment and is now a healthy tree).
I guess it all depends what is really happening in your cuttings, and what has really infected them. I agree with Jason that if it's "rot", then you're probably not going to save those cuttings. Depends on what bacteria/fungus/other pathology is occurring. If it's something else going on, then maybe... depends. I probably wouldn't go to great lengths unless they're cuttings that are hard to get or otherwise valuable to you for one reason or another. Good luck.
Mike central NY state, zone 5