The main conclusion of long-standing discussions concerning the role of solutions with degenerate metric (g ≡ det(g_μν) = 0 and even with g_μν = 0) was that in the first-order formalism they are physically acceptable and must be included in the path integral. In particular, they may describe topology changes and reduction of the 'metrical dimension' of spacetime. The latter implies disappearance of the volume element of a 4D spacetime in a neighborhood of the point with g = 0. We pay attention to the fact that besides , the 4D spacetime differentiable manifold also possesses a 'manifold volume measure' (MVM) described by a 4-form which is sign indefinite and generically independent of the metric. The first-order formalism proceeds with an originally independent connection and metric structures of the spacetime manifold. In this paper we bring up the question of whether the first-order formalism should be supplemented with degrees of freedom of the spacetime differentiable manifold itself, e.g. by means of the MVM. It turns out that adding the MVM degrees of freedom to the action principle in the first-order formalism one can realize very interesting dynamics. Such a two measures field theory (TMT) enables radically new approaches to the resolution of the cosmological constant problem. We show that fine tuning free solutions describing a transition to the Λ = 0 state involve oscillations of g_μν and MVM around zero. The latter can be treated as a dynamics involving changes of orientation of the spacetime manifold. As we have shown earlier, in realistic scale invariant models (SIM), solutions formulated in the Einstein frame satisfy all existing tests of general relativity (GR). Here we reveal surprisingly that in SIM, all ground-state solutions with Λ ≠ 0 appear to be degenerate either in g₀₀ or in MVM. Sign indefiniteness of MVM in a natural way yields a dynamical realization of a phantom cosmology (w < −1). It is very important that for all solutions, the metric tensor rewritten in the Einstein frame has regularity properties exactly as in GR. We discuss new physical effects which arise from this theory and in particular the strong gravity effect in high energy physics experiments.