Yup, you're confusing the values of the different config files. Basically, the tilde character (the ~ in ~andrewc/) is magical, and has little to do with the <Directory> defined in your andrewc.conf file. There are a couple of different ways of doing what you want, but the most understand is to use an "Alias", like so:
Alias /~andrewc/ "/Users/andrewc/Documents/SoftwareDev/Java/RMI/Trim"
You'd throw that in your andrewc.conf right above your <Directory> declaration (actually above or below - doesn't make a difference). You also want to make sure that your Documents directory, as well as all the subdirectories leading up to your Trim destination are set for 755 permissions. Otherwise, Apache won't be able to serve documents from that location.
I talk a bit about the "Alias" directive in the recently released Part 6, and will discuss the tilde character and it's magic in part 7 (not yet released).