tlippy, a 'cross bike could be an excellent choice for you. It will handle the road and the dirt/gravel with equal aplomb, and give you whatever degree of aero positioning for road riding that you desire. Some considerations:
 Gearing: Most 'cross bikes have double chainrings and cassettes of modest range. This will yield higher gearing -- the "low gears" will be higher than your MTB. Make sure the gearing isn't too high for you, and if it is discuss with the bike shop the possibility of swapping a triple crankset, and/or a wider-range cassette.
 Bar height: Modern threadless headsets make it difficult to raise the handlebars once the bike has been fitted and assembled. To the extent it can be done, it's accomplished by swapping out the stem for one with a steeper rise angle. If bar height is an issue -- if a bar position significantly lower than the saddle height is going to be a problem -- discuss this with the shop. A new bike usually comes from the factory with the steerer tube uncut, so the shop can wait for your fitting session to determine the height at which the cut should be, so you can have the bars positioned as high as you want them.
My feeling with drop bars is that you should set the height so that you can ride comfortably with your hands in the drops (otherwise, why have drop bars at all?). The tops of the bars will then just end up at whatever height they end up.
The Fuji and the Bianchi are both fine bikes. So's the Surly, and Trek XO1, and several others. Even my ti Airborne Carpe Diem can be equipped as a 'cross bike, although mine is set up as a tourer/commuter. 'Cross frames are quite versatile. Make sure it has rack braze-ons and eyelets if you ever might want to carry panniers and such.
You can, of course, put robust wheels and 'cross tires on other types of bikes, notably touring bikes. In fact, my touring bike actually came with Avocet CrossK tires, which work quite well as commuting tires and last forever.
Training: 2002 Fuji Roubaix Pro (105 triple)
Commuting/Daytripping: 2001 Airborne Carpe Diem (Ultegra/XTR, touring wheels)
Commuting/Touring: 2000 Novara Randonee (Sora/Tiagra/LX, fenders, lights)