It's a good rule of thumb with any rhizome/tuber/bulb that while it is dormant, it should get MINIMAL water. Most will rot fairly easy with too much.

If the majority of the rhizome is still firm, I might try and trim off the rotten area and stick them back in the pot. Sometimes rhizomes can be slow, but as long as they are still firm, all shouldn't be lost yet.

Also you mention temperature. I have no idea what the conditions are like where you are but; if it gets hot in the day and then drops substantially at night - it is going to take the soil a long time to warm up. It's constant warm soil, not daytime temps, that make them break dormancy.

Botyrtis blight and rhizome rot is seconded in my idea. When I divide, I keep the rhizome barely moist, and making sure that there are no soft regions with the root mass, as well as any apparent discoloring. Maybe incorporating some supplement of potassium to your watering schedule. But, that won't work or help if the rhizome has called it quits. My rhizomes, from division to finish, are grown in the hothouse, with humidity, watering, fertilizer, and growth all regulated. Wth this, bi-weekly inspections of the rhizome and fleshy root masses are done to ensure proper growth and healthy systems.

Thanks for your cheering words, minibim! I've done what you suggest -- cut off the dark bits, down to the crisp white part of the rhizomes and replanted them in some nice potting soil.
Temperatures here are in mid 25's to 30's, so if they don't sprout in the next 3 weeks or so, they never will. How much water should I give them -- should they be moist all the time, or dry out in between? (I try to show love with water, and tend to kill plants with it!)

30C is .... mid 80F so I would guess tempature is not the problem. Sounds like either, or both, your soil is holding too much moister for these ginger roots or you planted them too deep. Hedy. grow at or very near to the soils surface. Deep planting, particularly in a wet soil, is sure to promote rot.
So plant the roots shallowly and in a well drained soil. They do want moisture but not a soil that is boggy in any way.
Mike

I have a yard with many varieties of hedychium, which grow prolifically here in Florida. I generally start new rhizomes in a pot, with a mix of 50% vermiculite. I water weekly, but keep the pot in a water tray so they don't dry out. Once they sprout the first leaf, I transplant.
Keep in mind that hedychium make excellent container plants, suitable for patios, as they do well in filtered sunlight.