We have seen some problems on Azure relating to the database connection. IIRC, when I debugged it, simply running a query that touched nothing -- I think it was literally SELECT 1; -- would take around 2 seconds. The first query of a connection was fast, but then all other connections were slow.
I wasn't sure if it was specific to that user's Azure configuration, but it appears it may not be.
That could be related to friendly URLs being enabled.
Admin CP > Options > Search Engine Optimization (SEO) > Use Full Friendly URLs
Switch that off to see if it fixes the issue. If it does, but you want to keep friendly URLs, it will likely need some additional configuration for the new web server. Details here: Friendly URLs | XenForo
Now - I tried resetting the password on one of my accounts, but I never got the email. This concerns me, since it means new users may never get a confirmation email. Do you know what might cause this after a server move?
I have no idea when it comes to Azure, but even so we generally wouldn't make recommendations or support you with regards to server configuration (regardless of the type of server or its software). It's likely a better question asked to Azure Support. There may be some sort of built in solution - after all, email is a pretty core function for most mail servers, there should be something, somewhere.
Otherwise, you'd be looking to use an external SMTP server. Some people use Google Apps (or similar) for this.