For a lens with 18-200 mm range, I'd say that the Nikon 18-200 mm with VR is excellent, and it has VR to help with those 200 mm shots when you need it. There's going to be some distortion, but if you don't notice (because you don't know what it is), then you won't care; if you notice, you still may not care.

And besides, a bit of distortion won't hurt if you're not in a situation where there are a lot of straight lines, like in a city. It's almost a non-factor when photographing nature, landscapes, etc.
Anyway, I'd say it's a treat. I wouldn't want to switch lenses too often while I'm down there, and this is one way to avoid it. The lens is tops for such a lens.
For non-sports and action photography you could look at the 80-400 VR, however this is a very slow lens. It does have a good (and rather unique) focal length coverage. It's a popular wildlife lens, but the AF isn't very accurate (I've heard they're coming out with a new VRII / AF-S variant of this lens.) It's also rather expensive. Sigma has an OS (optical stabilization) version of this lens as well, but it suffers a similar fate - it's price just represents this a little better.
Isn't the Sigma 80-400 mm OS lens also HSM, or am I thinking about the Bigma? Haha, it's so weird, because the 80-400 mm has OS but apparently no HSM, while the Bigma 50-500 mm has no OS, but includes HSM. One day, Sigma are going to do it right and include HSM on the 80-400 mm, and OS on the 50-500 mm, because at the focal lengths, and focal range you're talking about with these lenses, you really need both. Of course, OS isn't going to be a replacement for a monopod or tripod at 400 or 500 mm, and who'd want to run around all day with a big lens like that? I can sort of see why they didn't include OS in the 50-500 mm, but anyway, I'd still like to see it.
I'd love to have either lens, although I have heard better things about the Bigma.