We outline a geometric method of constructing generic off-diagonal and diagonal cosmological solutions of effective Einstein equations modeling modified gravity theories with two non-Riemannian volume forms and associated bimetric and/or biconnection geometric structures. Such solutions are determined by generating functions, effective sources and integration constants and characterized by nonholonomic frame torsion effects. In the physical Einstein frame, the constructions involve: (i) nonlinear re-parametrization symmetries of the generating functions and effective sources; (ii) effective potentials for the scalar field with possible two flat regions which allows unified description of locally anisotropic and/or isotropic early universe inflation related to acceleration cosmology and dark energy; (iii) there are “emergent universes” described by off-diagonal and diagonal solutions for certain nonholonomic phases and parametric cosmological evolution resulting in various inflationary phases; (iv) we can reproduce in two measure theories massive gravity effects.