Time Machine backup fails

Several weeks ago I got repeated messages on my iMac saying that the TM backup had failed, and indeed the last previous backup was usually at least several hours earlier. I was using a G-drive mini that had worked flawlessly for years. In an email exchange with the G-drive people, I was told that it was probably time to replace the mini. So I bought a new G-drive (I think mini is no longer produced), and it seemed to work fine.

But--it too is now repeatedly unable to do a backup. One issue: our wi-fi has been a bit irregular of late, running very slowly at times, and requiring a modem unplug and reboot to get it going. The modem is fairly new, and I can't find anything on CenturyLink's website to diagnose its health. I know I have to set a day aside to call them and see what they might be able to figure out.

A couple of additional factors: a few months ago, my not-quite-year-old iMac just quit. I don't remember exactly, but I think I did a restart for something, and it just wouldn't load and reboot. I spent more than an hour on the phone with an Apple guy, who was terrific, and ultimately I ended up with new system stuff (I think--I'm out of my depth here). One result is that I occasionally get security checks (like just now, from Apple) because I'm using an "unknown device." Is it possible that I've still got some weird glitch buried in my computer?

And, oddly, the little icon for TM on the upper right of the desktop doesn't spin when a backup is in progress--I'm pretty sure it used to do that.

Meanwhile I'm hoping someone can suggest a fix that doesn't involve what will be an excruciating exchange with our phone company/ISP.

And maybe someone can even verify my experience with this computer as not so surprising: I have had more troubles and glitches with this computer (mfg. "late 2015) and Sierra than I have with all of our previous Macs, dating to 1987(?) put together! Even my magic mouse is on the fritz, but that's a different forum . . .