It seems that if I unmount a web module (that was deployed as root) and then mount another web
module and try to run it (also deployed as root), Tomcat continues to try to execute the previous
web module.

For example, I create a simple web module that has index.jsp forwarding
to a servlet.  It runs fine.  Then I unmount it, and try to run another
web module that also has an index.jsp in it--in this case it's the
struts example application.  This second app fails because Tomcat is
still looking for the original servlet from the first app.

This is a common situation when developing web modules.  At first, the web module is tested within
the IDE and it's deployed to the Internal tomcat using the root context.  It's not until later that the
module may be assigned a different context.

Given the situation, the only "solution" is to shutdown the IDE, delete
it, and then reinstall it.