Each virtual machine is backed by its own vhd, which resides in a Page Blob in Azure Storage. This is not a good place to store any type of content that needs to be shared across multiple instances (and in-proc session state will also be limited to the running instance of that virtual machine). You need to store your session state and any other persisted data in a common place accessible to both Virtual Machines, but do not store content directly on the vhd's main disk, or the temporary disk.
For session state, you can easily set up Windows Azure SQL Database (basically sql-as-a-service) and use that as your session state provider. Or you could run memcached, use Table Storage, or choose something entirely different - I'll leave other ideas to your imagination.
Same goes for running WordPress - content would need to be in a common data store. Since WordPress uses MySQL, you can set up MySQL in an independent virtual machin instance, or subscribe to a MySQL database via the Azure Store.
One other option is to store content in Blob storage, which is completely independent of any virtual machines you create, and very easy to access. Blob storage, along with Table storage, has excellent language SDK support that's built atop the REST API (currently .net, php, java, python, node, and ruby SDKs). You can grab the SDK of choice from the Azure download page. You can also peruse the source code, in the github repo.