OK, I did the needful and contacted Audible. After only a short wait, I got a helpful person who explained that "yes, they _are_ having problems, but there's a workaround". Turns out the workaround is to enable Software Verification for the Account. So:
- Log on to Audible
- From the dropdown list under your name, select Account Details
- Re-enter password if required
- In Your Account, click Update Settings
- Check the Software Verification box.
- Proceed to your Library to download books
They claim this is temporary. I sure hope so. I do NOT want them checking my desktop, thank you.