The phones mercifully finally stopped insisting there was voicemail that same evening. I actually had already planned to contact Panasonic about it the next day as well, but that wasn't necessary. Apparently clearing that value reset everything, at least for the moment, but there was still a delay before the effect was apparent.
Thank you for mentioning that procedure, though it didn't sound like you necessarily expected that to work around it.
I still don't understand how it got into that state in the first place, though; that value is an active one that represents the last result of the Obi110 connecting with Google Voice and checking for voicemail, correct? If so, how can it get "stuck" like that? The only condition I could anticipate is an inability to connect to Google during that entire weeks-long period, but since I've been placing and receiving calls and voicemail through it that condition doesn't seem possible. What else could get the Obi "stuck"? Until I know what caused it and how to prevent it, it may happen again after more weeks of voicemail and calls. I know how to work around it, but I'd like to prevent it.