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Degeneracy is a method to accommodate exact, low energy vacuum states in scalar-tensor gravitational models despite the presence of an arbitrarily large vacuum energy. However, this approach requires very particular combinations of scalar field and metric couplings in the Lagrangian. In this work we study departures from the restrictive degeneracy condition -- starting from a fiducial model containing an exact Minkowski space solution, we break the degeneracy condition in numerous simple ways to test if the resulting models maintain certain key features -- specifically the dynamical cancellation of a large vacuum energy by the scalar field and the existence of a low energy vacuum state. We highlight the role the tadpole plays in eliminating the fixed points of the dynamical system, generically rendering both the scalar field and metric time dependent. Our results indicate that when violating the degeneracy condition but preserving shift symmetry, the metric maintains an asymptotic Minkowski state, irrespective of the presence of the cosmological constant. In contrast, when shift symmetry is also broken the asymptotic behaviour can radically alter. Regardless, the non-degenerate models in this work share an attractive quality; harboring low energy, late-time asymptotic states that are independent of the vacuum energy. The tadpole allows for a broader class of non-degenerate, self-tuning models than was previously realized.