Here's what happened and what I discovered. It may help someone else, some time.
Internet via Motorola Canopy radio looking at one of my ISP's nodes on a line-of-sight tower. Wireless router. A PC is wired to the router. A Chromebook and two Droids (Razr Maxx HD, but it doesn't matter much) use the WIFI connection. PC had good, working Internet access. All wireless devices showed excellent WIFI signals and connected. You could see the WIFI indicator fire as attempts were made to access web sites. Droid accessed Ingtyernet properly when forced to use Verizon. Thus, the title of this post. When I thought it was just one Droid affected, went through all possibilities (there aren't many) with Verizon tech support. Concluded that a factory reset was all that was left and planned that for the next day.
But, the next day, I discovered that all three wireless devices were affected and that it was only some sites and some parts of sites, while others were unaffected. The clue was that the Chromebook responded to a failure with FAILED LOOKUP. The problem apparently lay with the difference between the Internet connection definition structures of a Windows PC, with it versatility in configuring connections, and the way the simpler operating systems do it. The upshot was that the PC's queries were being routed to one DNS server, while the others were being routed to one with a corrupt DNS table. The fact that the Chromebook had the problem even when it was on a wired connection shows it had nothing to do with the devices being wireless and nothing to do with WIFI, per se.
It was nothing my ISP's remote tech support contractor could deal with, and about the middle of the next morning, before I could call the ISP home tech people, it was fixed, and all devices were fine. The lesson is that it can only look like a phone is somehow faulty in its WIFI Internet access and a PC is working fine, when it's actually some glitch or malconfiguration at the ISP's equipment or in some servers they use. Nothing that can go wrong with an Android can make a second Android and Chromebook all fail Internet access over a working WIFI connection. It's an ISP problem. Bring in some other devices and see if they work and/or try your device on another open WIFI.
If Verizon Tech sees this, you can look up the incident in the database under my primary phone number and add the appropriate note for your knowledgebase.
By the way, FANTASTIC and very patient (like 2 hours) tech support guy last night