'91 Olds Toronado Trofeo won't start

It has a fully charged battery, and a new starter. Engine only has 120k
miles on it, and previous owner kept very good care of the car. No rust
on bottom, trans fluid is nice and bright red, all other fluids including oil
check out fine.

It drove just fine a few weeks ago when I bought it, but now when I turn
the key it just clicks once and doesn't actually fire up. I left it sitting for a
week, and it started one time, but hasn't been able to work since.

The starter is very new and was installed on the car before I bought it.

Is it possible that the new starter is faulty in some way?

If it was a problem with the anti-theft system, would the starter even
click?

I'm kind of at the end of my options here, I've been dilligantly doing lots
of research on the potential problems this week but haven't been able to
figure out the problem.

If it was the spark plugs, I assume that the symptoms would be different
than just a starter click.

Any ideas?