Characters currently live on any of the closing servers can be transferred for free, and there's a guide to doing soThere's no time limit for switching servers.
It's already been a good run for Lord of the Rings Online, which opened its doors way backbut the closures don't necessarily mean that the game's winding down. The smaller number of severs should ensure a more tight-knit community and really I don't imagine that the game costs a huge amount to run at this point. There's some life in the old dog yet.
The remaining US servers are Arkenstone, Crickhollow, Landroval, Gladden, and Brandywine; the European ones are: Belegaer, Gwaihir, Sirannon, Laurelin, and Evernight.