Moving to a per-tenant database sounds like even more work to me than moving to shards. Moving to per-tenant means rewriting _everything_ - moving to shards has you rewriting a lot less.
There are logical DBs (folders/files on a single physical server), and there's a few physical servers. We're currently at the stage where the first physical server hosts most of smaller organizations, and all other physical servers are usually dedicated servers for larger clients with high loads.