While the racers might shudder....cyclo cross bike. The tires are a little fatter, but it is still pretty much road bike is geometry with drop bars.
I toyed with the cross solution and ended up building up a Surly Pacer frame (road frame, not cross, but with room for fatter tires). With fenders I can run 28mm which gets me through winter slush, grit and most limestone trails. If I was going to ride the limestone trails more, I'd probably switch to 32 or 35 for the summer and take the fenders off.
Oh, and welcome to the boards!