I called tech support and told the tech what I was dealing with. He had me click through and check settings, then he came to the conclusion that NOTHING was wrong.
The settings might be correct, but there was definitely something wrong. I was already set to look for another provider on the 1st of the month.
Today, I waited 10 minutes and gave up. I called tech support back. I got lucky and received a knowledgeable person. he asked me about splitters and such on my line, then had me add an extra string to my modem setup. The result actually capped the connection speed to 36,000, but the slower connection speed actually made my surfing faster than before. That is because the amount of lost packets decreased drastically. Now that my computer doesn't have to resend packets, I am able to surf faster.
I should have gotten the first tech's name, so I could call his supervisor and explain the situation. They almost lost a customer. I don't know if he was at the end of a long shift and tired or just having a bad day, but he didn't seem too interested in helping me at all.
I might have run into the same trouble with the next provider. I don't know if the problem is due to the line quality of my phone line. I can't hear any static or ambient noise on my phone, but the quality could have degraded just enough to start causing the connection issues.
Regardless of the problem, I am able to connect to the Internet again on the first try and I can see a noticeable speed increase even though my overall connection speed has been capped at a lower speed.