A lot of games have been doing this. And i really have no idea why SE isnt doing this themselves.
Have a separate Character Server.
That way you first log in to a "Character", and then independantly to that select which Server/World you like to play on.
That way, if one server is full, you can still play your character on another server for a bit till the server of your preference gets a free slot.
Naturally there should be something of a "Home" server your character originally got created on, but atleast it would fix congestion a little by spreading out people who arent looking to raid with their Free Company at the moment but just want to play the game. Also means a LOT less alternative characters that have to be created on other servers. SE has already shown that Server-Crossing battle instances are perfectly possible, so why not.
Yes, you wouldnt be able to chat with your linkshell (plus the pearls have to be server bound) but that's a small price to pay.
Also, why i brought up the "home" server. If there currently is a character bound (i.e. originally created their character with the same name on that server) to the server you are trying to join temporarily with a similar name, it should refuse the connection and you will simply have to select another one. Easy fix, and that way people always have atleast one server to join. Will it suck a little for the hundreds of Sephiroths, Clouds and popular anime names? maybe. But if you would have only been original, heck... no problems at all.
Worst case, you could always log in on one of the many other servers SE is going to errect without having to completely start over from scratch.