Well, things seem to be working at the moment, and I think it was WeeJeem's suggestion......
Firstly I discovered that I couldn't ping all addresses from the command line - google and yahoo would work but others, like pprune, wouldn't. So I made the change WeeJeem suggested, rebooted - and it still didn't work. I could get google up immediately after the reboot, but then nothing. All the time other programs could connect fine - google drive, Norton, updating malware defs - they all worked fine. It was just the browsers.
Anyway, I disabled the firewall in Norton and it started working - cracked it, I thought. So I went back to the original DNS settings, switched the firewall off - still didn't work. Back to the WeeJeem settings, firewall off and it worked.
Mmmm, I thought, strange. So rebooted the machine, firewall was now on and WeeJeem settings in place, and it worked. And has continued to work.
So I think those settings are what's made the difference, but I couldn't get consistent results so I'm not 100% sure. They certainly didn't work the first time I tried them, and I'm not aware that I've set anything else differently. So let's see what happens......
Thanks to all who posted.