@Scott: welcome to Java Ranch!
@Adam: I have the same version of Eclipse, and did the same step as you did, and here is my web.xml:
I have no idea why it isn't working for you. You might check the log file (workspace/.metadata/.log) to see if there is any clue as to what went wrong.
Some other things to try:
a) Create a new workspace and a new dynamic web project, and a servlet, there.
b) Install another copy of Eclipse in another location and run it; have it create a new workspace and try again with the dynamic web project and servlet.
If "a" works, then your workspace/.metadata is screwed up. If "b" works then you Eclipse installation is screwed up. In either case you can copy your project directories from the old workspace to the new workspace and use File | Import | Existing Project to get them into your new workspace (and thus not loose any work).