Transit and eclipse spectroscopy offer insights into the chemical and radiative properties of extrasolar planets. Inferring said properties from their spectra necessarily requires data-model comparisons. Typically, Bayesian atmospheric retrieval methods have been used to derive these properties. Retrievals generally make minimal assumptions allowing for more flexibility in the fits and potentially illuminating unknown physics, but this also creates the possibility of unphysical atmospheric determinations. In this work, we combine powerful Bayesian parameter estimation tools with a new grid of self-consistent, one dimensional radiative-convective equilibrium models. Using this framework, we determine metallicities, carbon-to-oxygen ratios, and heat redistributions for a population of hot Jupiters from their dayside thermal emission spectra. Our sample is derived from a uniform data-reduction analysis of spatial scan Hubble Space Telescope WFC3 and Spitzer 3.6/4.5 micron observations sourced from a variety of programs. The self-consistent approach mitigates the possibility of unphysical solutions, resulting in robust atmospheric inferences. We will present our self-consistent constraints on these fundamental atmospheric quantities in the context of current retrieval results and compare to atmospheric composition predictions from planet formation theories. Our nominal model fits to each object will be extrapolated to the James Webb Space Telescope wavelengths, as well as high-resolution, cross-correlation instruments, in order to identify observational setups that best address any current degeneracies.