Connection issues during a drop in CPU usage
It sounds somewhat like you are hitting a bit of memory pressure. Dirty Buffers are pages in memory that have changes in them that have not been written to disk yet. Normally they would stay in memory until one of the various processes clean them up and/or write them to disk.

There are some things you will want to look into but with only the information you have given here I have to ask, "How much ram have you reserved for the OS?"

You are running a 64 bit version of Windows. Now, the 32 bit kernel will use 4 GB of RAM. However, we have found (the hard way) that 64 bit versions of the OS will use 8 - 12 GB of RAM dependent on how busy the Sql server is. By that I mean are you doing a lot of reading and writing to disk. If the OS can't allocate the memory it needs for this (because the Sql server has sucked it all up) you will see the CPU being consumed for the additional context switching it needs to do to write those buffers out to disk. When you added the additional RAM to the server did you reset the Sql Server Max Memory setting? When you only had 16 GB of RAM your Sql server was constrained as much as the windows kernel was and so you were likely using virtual disk (hard drive space) quite a bit. When you added more RAM you opened things up a bit and now the Sql server is attempting to keep more of the database(s) in memory.