The other modem may have something to do with the problem. If it's SBC/Yahoo then you're connecting by PPPoE, right? If that's the case, then you'll need the Roaring Penguin package (rp-pppoe-3.5-i386-1.tgz in the slackware/n/ directory). When you set this up, you're /etc/ppp/options file must be empty, so the settings that you have in there for your modem have to go. Copy the one you have now, then 'touch' a new one.
Check this thread:
If you aren't with SBC/Yahoo or you aren't using PPPoE, then try to give us some more info about your DSL connection. Read through the literature that came with the package and tell us whatever you can.