For me, the problem was that IMs I sent to other people would come across double to them, although I only saw one. When I posted to local Chat, it would come across double for me as well. It may have affected payments. I experimented by purchasing a 1 linden item. My account balance momentarily dropped by 2 lindens, then snapped back up to only one linden down. I credit the sophistication of the vendor for that.
I tried logging out of SL, disabling and re-enabling the network adapter, shutting down the modem and restarting that, re-installing to the latest Firestorm browser version, and fiddling with the settings in Preferences, all with no effect. What finally worked was just restarting the computer itself. If other people having this issue are like me, and are resistant to closing everything they have open and simply leave their computers in sleep mode for months at a time, they might want to start early with the tried and true software repair technique of rebooting the computer.
Thanks also to Alwin Alcott for the more detailed advice given at