I finally disconnected the phone that was directly connected to the landline, which means Asterisk now decides when the phone rings at all and any voicemails get mailed to me as WAV files. This is much better.
I did have to tweak the voicemail settings so the WAV files were actually audible without turning the PC speakers waaay up. In /etc/asterisk/voicemail.conf, I set volgain to 8.0 (and installed sox).
Asterisk has “taken care of” 358 calls (callers with no CID or CID identifying them as toll free numbers) since I set it up early last month. They went straight to voice mail without ever ringing my phone. Very few of them left messages.
(Yes, I still plan to post something more descriptive with configs describing as much of my setup as seems reasonable.)