How do I set up Dropbox API for use on a multiple server environment/ load balancing?

I'm still not clear what change you made exactly regarding the cookies and sessions, but that certainly would be relevant here, as the error is indicating that the CSRF token didn't match what was expected, and the SDK stores the expected CSRF token in the session. You can see that here in the SDK where the exception is thrown:

https://github.com/dropbox/dropbox-sdk-php/blob/9db8328c6c035baa0fee48291bd8bd93bcc714b0/lib/Dropbox/WebAuth.php#L246

We don't have any specific documentation on how to handle this in an environment like yours, and in any case I suspect it would be specific to the environment, so I recommend looking at how the code works, and updating the your code and/or the SDK as necessary.