I am no expert on this and don´t have any concrete solutions and also the tips here are possibly not even applicable in your case but thought I´d share what I would look at. I know there are different scripts or snippets of code one can use on HTML and other web pages that instruct the browser to refresh or disable the cache etc. and something like that might trigger the browser to do a reload automatically which then maybe direct the user to the new server. The question is, at which "touchpoint" can/should you have that script/code and also related, where is the "hickup" i.e what part of you infrastructure is it that cause this because obviously that is where you should target your "fix".
Also read up on DNS flushing, that might be related or be something you can use? I can´t remember exactly what it does but think it is related to exactly these types of issues.
Another thing you could do is create some redirect somewhere or even in Tomcat on server A IF the user actually gets there (or are they stuck in some limbo in between server A and B possibly) or DNS trick in between so domain > another domain > server.
PS: If nothing else, try googling "dns propagation problems" where you have a lot of information about similar problems and I guess also solutions.