What happens with the external modem, well, first I should say that
Linux will automatically boot up and dial-in to my ISP as I am logging
on the computer. That is normal. I hear the dial tones and the
buzzing sounds, like the dial-ups make, when connecting to the internet.

The dial tone comes on for a second, and it acts like it wants to
dial, then it drops the connection before it follows up with the
numbers to dial-in to my ISP.

It doesn't handshake because I don't get that far into the log-on
process. The signal is dropped after the first number or two is dialed.

I've tried to go into Network settings but so far no luck.