On the third call, the person connected me to a next tier person. Whatever test she did on my OOMA line showed no problem. She had me test my internet connection using speedtest.ph from my laptop - not one of the standard speed test sites.
My laptop was connected via wifi. It showed good speed up and down, and 5/5 for Skype type connections. It also showed a jitter of 9 ms. She jumped on that as the cause. Given the exact problem I've described in the OP, that made no sense to me. Kind of like "when was the last time you defragged your hard drive?"
She said I had to have a jitter of 5 ms or less for OOMA to work properly. First of all, I doubt my jitter had changed since a month or two ago when OOMA was fine.
After talking with my ISP, I did another speed test. This time instead of using my laptop over wifi, I used an ethernet cable plugged in to the same router as the OOMA. This time my laptop measured a jitter of 4-5 ms. So the OOMA must also have that same amount of jitter but still has the echo/dropout problem.
Back to square one.