O.K., I think mine is at least in part a hardware issue. I changed out to my other extended battery back, and I pressed on the back where the antennas are (top right hand side of the phone as you face the phone), got a 4G/LTE symbol, then ran a speed test - almost 22 down, over 4 up. If I squeeze and let up on the top corner of the phone, I can see the speed fluctuate more widely than usual in a speed test. Time for a call, and maybe an Asurion claim. I can't figure out what the root cause is, because the phone has not been seriously dropped, stepped on, etc.
EDIT: I "tweaked" the antenna connectors on the phone, getting them out a little more. Seems to have made a difference in the reliability of the connections.