I started a similar thread a few months ago, but I was on the fence between the FA* 600 and the 560. In the end, the 560 won and I'm glad it did. I couldn't imagine my 3 hour hikes with the 15 lb 600 + camera body + grip + heavy duty tripod + gimbal head. Sharpness from the 560 is unbelievable wide open, colours are insane, rich, vibrant. I've heard a few people mention that the sigma is smaller, therefore hand holdable, but at that weight I'm not sure you'd really be able to hold it for long while tracking birds.
While it may take longer to find a buyer to drop $$ for the 560 if you choose to sell it later, it'll retain it's value more than an off brand. The downside that no one even brought up when I was looking, the 560 is exactly the length for carry on baggage. If you plan to travel with it, be prepared to either part with it while it goes off in to nowhere land under the plane or put up a stink about the 1" it's over while nestled in it's case.
I shoot tons of shorebirds and absolutely love using the 560. If you can make it up to Seattle, Glazers has had the 560 in.