Think I finally figured it out. I didn't understand why the Avast Management Service wouldn't stay running. I thought for a little while that it couldn't talk to the database. I checked permissions, changed permissions, compared everything that I could think of with my working SBS 2008 box, and I couldn't find anything wrong.
Finally, I looked at the event viewer, under "Applications and Services Logs" > Anti-virus.
I kept getting
"Error 0000A410 has occured while opening license file D:\Program Files\Alwil Software\Management Tools\DATA\license.dat. Is the license file correct?"
errors.
So, I checked permissions on that file, which seemed ok. So I renamed it, and made that change in the AMS Maintenance tool, under "License." It still didn't work, so I renamed it back. Then I checked the database connection which was fine.
Then, I went back to SQL Connection, put in the server info:
Sql Server Instance: server\avast
Database: avast
login: sa
password: "password that I setup during SQL setup"
When I hit ok, and it said that it had to restart the service, it gave me an error: 5, so I restarted the service manually.
I then noticed that it wasn't shutting down anymore by itself.
So, I tried to open the ADNM console, and tried to have it detect servers. It still didn't do that, so I typed in the server's full netbios name, typed in the username and default password (admin) and she came up!
Now, as soon as I got it up and running, I came running here so that I could get the information out there. Hopefully it stays running through a reboot or two, and we can close this unfortunate case for good. I'll keep you all posted!