I'm the original poster. I didn't expect THIS much response - thanks guys. There are some suggestions here that I'm going to try. As to mirrors, I went thru about 4-5 before settling on the handlebar one I have now. It is piss-poor compared to the excellent mirrors on cars. I can barely trust it to watch for overtaking bike traffic, let alone cars. I did try the helmet mirror but couldn't make it work and gave up on it.

No I don't want to ride with no hands (though it would be a nice show-off trick for an old man). What I want to do is turn and survey traffic without losing balance and veering off course. That's why I brought no-hand riding into the question, it probably shows my lack of balance. And yes, my body-swivel capability is restricted by age.