I'm on a similar quest and have a tentative answer (I'm currently working it out). My definition of a generalized differential manifold would be: A locally ringed space with nontrivial cotangent sheaf. Covariant derivatives can be defined using only the cotangent sheaf, and there is a dual version of the fundamental lemma of Riemannian geometry for this. (My creed: forget tangent space.)
The abstract differential geometry of Mallios et al looks quite attractive to me, but I haven't yet read it in detail (lacking time, money, and a useable math library). One immediate problem I see with their definition of vector bundle, being the usual suspect, a locally finite and free sheaf module. But a major motivation for using sheaves is infinite dimensional applications, like path space.
I have an incomplete copy of yummy lectures by Jürgen Bingener, Regensburg 1983/4 on basics of calculus and Riemannian geometry in the language of ringed spaces. Need to contact him and ask for more.