I have a simple hello, world servlet application that I am just playing around with, and pushing it out to my tomcat server on a VPS.
When I make a change to my code
Sounds like your class loader is not loading the servlet classes once they are updated. This might be fixed if you change your web.xml file which should prompt the server/container to re-deploy and reload the servlet classes. I guess add an empty line at the end of your web.xml and save it and then see if that fixes it. As i said this might fix it or might not.
Good luck!