A very simple rule of thumb is "Don't use instance variables in servlets".
When in doubt, create your variables locally, even if it makes the service methods (doPost, doGet..) longer.
Most larger, well written projects move toward a front controller or command
pattern to break up and organize the code; which takes care of the long service methods.
If you start synchronizing access to instance variables, you're likely to hurt the performance of your application by causing requests to queue up instead of running concurrently and independently of one another.