If you are adept at testing all parameters...

I think the first test is to disconnect the modem from the computer,
power off the modem, power it on, and see if it will sync in a minute or
two.

I am not adept at all but during the last few weeks, as my connection
was getting weaker and weaker, I learned all these steps. For a while
it did work. But now nothing works: rebooting the modem, disconnecting
and reconnecting, rebooting the computer, some combination of these.
Only lights go on and off forever.

The diagnosis that modem is dead was by AT's tech, although they never
came here.

Dial up does work, thank God I have an older computer so it does have
an internal dialup modem. That's what I am using right now but this is
extremely unpopular with family as it ties up the phone, one reason why
got the dsl in the first place.