Spoke with Apple Support. No grins. Here's where things stand.
1. They are aware of the problem and are looking into a solution.
2. They asked that I do a full Restore to isolate whether it's a hardware of software issue. I explained that since this problem affects both 5S and 6 phones, and that since the problem didn't occur until the update was released, that pretty much puts the focus on the IOS update. They agreed.
3. They recommended that I don't listen to podcasts in my headset until they come up with a solution, and that if I need to listen to podcasts while driving, find another way to listen.
4. They'll let us know when they have a solution.
I also said that I don't mind passing along what I learn but that I really didn't intend to be an Apple spokesperson, so would they post something on this forum addressing the problem? They said... well, nothing, really. Not no, yes or maybe. So this is probably the only time you'll hear what Apple has to say.