Update: I connected the replacement (refurbished) H3. The settings and timers did not restore from the remote to the H3, as expected. I can deal with that, and it looks like I may have needed to recreate the timers anyway (guide changes). I did not try to copy the recordings over. I could not get it to activate from the H3 itself. I also couldn't log in to Dish on my PC to do it. Are they having problems, or am I singled out for some reason? Anyway, I called in and got the auto-attendant to activate it.
The Good: The original problem with the H3 is completely solved.
The Bad: It will not link with my Joey, so now I can't watch anything from it. It says that coms are good, but it is unlinked. The Joey does see the H3, because it does give the new number on the info screen. It also says it is correctly connected to the internet via the H3 through the Moca. I tried the suggested fix of resetting both the H3 and the Joey. I guess I will be calling support tonight and trying to resolve this.
The Annoying: My harmony hub/remote was paired to the old H3 via IP. When I try to pair it to the new one, the harmony app crashes, both on my PC and on my phone. The hub and remotes are no longer supported, so I don't expect help from Logitech. I think I can deal with this by switching to IR control, but that means deleting and re-adding the H3 as a device, which means rebuilding my remote buttons and activities. It's a pain, but doable (assuming the harmony software still works for that).
All in all, as my wife said, this is making it easier and easier to just jump to streaming.
Update: My dish login is working now. Interestingly, it says that the H3 is on, but not connected. It says that the Joey is on and internet connected. Weird...