Will a tilt-shift lens solve my ‘leaning building’ shots?

I shoot primarily with a Nikon D300, and that’s not likely to change soon. The camera does everything I need it to do, but this. I also use a 17–55 mm lens, which I know that Thom castigates as being too expensive for not enough lens, but it’s worked well for me for years.

I’m on a trip in Italy at the moment, though, and having some serious problems with looking up at large buildings going to infinity. Consider il Duomo di Milano (Milan Cathedral):

Il Duomo

It doesn’t actually tilt inward like that, it’s a pretty solidly built structure. For some reason, I’m really noticing on this trip that my building shots are just mangled.

Is this readily solved in post production such that the resulting image is printable and usable for something other than internet slideshows? Or should I be looking at a tilt-shift lens for this? And if I do look at the lens, I’ve been told that the flash overhang on the D300 means that the Nikon 24mm lens won’t fit — is that true?