Well, I've finally got past this problem and thought I would post what I did just in case in helps anyone else.
I'm not completely sure, but I think the base problem was the version of Java running on my Mac. I had the update for Java which gave me 1.6.0_17 as is needed by this version of Eclipse, but something wasn't set up correctly. When I did "java -version" at the terminal, it would tell me I was running the right version, but when I ran a test program from a browser window, it said I was still running 1.5.x. I even went to the Genius Bar at the Apple store and they couldn't figure out why it wasn't picking up 1.6, but I ended up just upgrading to Mac OS X 10.6.2 which runs the correct version of Java by default (I was running 10.5.8 previously).
After I upgraded, things still didn't work, so I completely deleted the GlassFish Tools bundle and Flash Builder 4 and reinstalled everything from scratch. I think it did change something because things seemed to operate slightly differently (e.g., the GlassFish v3 server was available by default and didn't have to be downloaded). Still, when I tried to run my little test app, it still couldn't access the database.
Finally, I randomly realized that I had the EntityManager injection located in a service file rather than in the bean file directly and once I moved it to the bean, the EntityManager variable was injected correctly. I'm not sure which of these things actually caused the problem or if it was a combination of all of them, but it finally happens correctly.
The last issue I had to get past was figuring out where the mysql-connector-java-5.1.7-bin.jar file should be. I put it in ~workspace/.metadata/.plugins/com.sun.enterprise.jst.server.sunappsrv92/domain1/lib/ext/ and restarted the server and was then finally able to access the database!
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