I agree with most of what was said here - based on your situation and what you say you shoot, I think it would be a good upgrade, especially if you can make the money work out relatively neutral. If it were me, I'd also trade the 12-24 DX as well. A couple points:
1. You will see at least 3 stops of high ISO performance over your D300 (which is at least 1.5 stops worse than the industry DX standard D7000 these days). I expect the D600 to be at least 1.5 stops better than the D7000. That said, if you are not shooting action indoors, with your fast lenses, I'm not sure why you're at ISO 3200 and above so much? You must be shooting in dim restaurants or caves.
2. I found that when I shot film (FX), there was a noticeable difference between 18mm and 20mm at the wide end. That is something you can rectify down the road, though. I have the VERY underappreciated 18-35 ED variable aperture ultrawide, which sells used pretty cheap, and still sells new, I think. It was a great lens on film, and it's a very inexpensive way to get to 18mm on FX digital.
3. I support your upgrade, mainly because I only see three reasons to shoot DX these days: 1) You shoot sports and/or wildlife, and can't afford the super expensive long pro glass to get close enough (my reason); 2) You want to use smaller, lighter lenses; 3) You just can't afford the entry price into FX DLSR's and better FX lenses.
Beyond those reasons, FX is superior in every way. Now that there is an FX body no bigger than a D7000, that eliminates the fourth reason: camera size/weight. I expect that the D600 won't be the least expensive FX body Nikon ever produces. My expectation is that demand will drive the development of a $1,200-1,500 FX body in the next few years. But,...I wouldn't put off a purchase now in anticipation of it.
I plan to buy a D600 next year, if the finances allow. I shoot a lot of basketball, where reach is not an issue, and low light is. I will always shoot DX, as well, because of outdoor sports and because for snap shots and light travel, the lenses are smaller. But, I am pretty sure that once I have a D600, the above applications will be the only time my D7000 sees daylight.