I've had problems in the game on my high-population server (Argent Dawn), but nothing game-breaking: the worst I've had is minute-long lag when I'm at the auction house at peak hours. It's frustrating, but it sure doesn't stop me from playing it obsessively.
I'm hoping they allow for character migration soon: this seems to me as if it'd be the easiest fix. From my little bit of DB knowledge, this would entail moving one record from a character table, up to 51 records from a talent table, up to a couple hundred records from an items table, up to five items from a skills table, up to a hundred records from a "learned recipes" table, up to two items from a mount table, and then possibly a handful of records from a handful of other tables. It doesn't seem as if it'd be too difficult to design a button on the character screen that allowed for this export/import to occur.
Daniel