One option is to create an abstract parent servlet class that you can extend, then expose the logic for handling exceptional behavior in a protected method or variables, which you can access from your public servlets.
You can also add something like this to you web.xml:
<error-page>
<exception-type>java.lang.Exception</exception-type >
<location>/ErrorHandler</location>
</error-page>
We use that in production for small projects with JSF and will do the work for Servlet and JSP. More info here.
What I like to do is hide the exceptions in production, but in development we just display it so it kinda speeds up development.
Just remmember that for scaling well with SOAP and REST you might have to do some extra work, but I think for your scenario it should be fine.