Then I posted a desperate cry for help on two forums (Stargazers Lounge and Cloudy Nights), emailed Tim Crawford (he's a mentor for the AAVSO), and emailed Paul Rodman, the creator of the AstroPlanner software I wanted to use.
Tim didn't have the answer, but he contacted other AAVSO members to ask if they had a solution. Similarly, Paul did not have the answer, but he posted my problem on the AstroPlanner Yahoo Group site.
The response from all of these sources was overwhelming. It's hard to diagnose this type of problem over the internet, but I got dozens of ideas from a lot of people.
As an aside, it is exciting to find how "connected" we are as amateur astronomers. I don't know anyone locally involved in astronomy, but I suddenly had lots of smart people helping me solve my problem, from around the world.
Anyway, back to the issue at hand. It was suggested that either Windows 7 was the problem, or my Meade USB/Serial adapter