Unfortunately, the handles of my patent D-bags have not softened with use, but I dont find them uncomfortable. My non-patent leather bags handles have softened a bit, though, so perhaps a non-patent D-bag might be a better choice for you?
When I travel with a leather bag, I use a folding/expandable nylon tote for protection (e.g., Longchamp, Herve Chapelier, Baggallini, etc.) when I place it in the security-check bin and under the seat. I would worry that a fabric dustbag wouldnt provide enough protection if the airplane floor or overhead bin is wet or dirty.
Also, sometimes I need an extra bag for the return trip, so the expandable nylon bags come in handy. I have been able to fit the D-bag under the seat with the handles folded down, so it shouldnt be a problem, and I wouldnt place the bag in the overhead bin, as someone might come along and put something on top of it that would ruin the bag (scratching or crushing). Hope this helps!