This paper presents cosmological results from the final data release of the WiggleZ Dark Energy Survey. We perform full analyses of different cosmological models using the WiggleZ power spectra measured at z = 0.22, 0.41, 0.60, and 0.78, combined with other cosmological data sets. The limiting factor in this analysis is the theoretical modeling of the galaxy power spectrum, including nonlinearities, galaxy bias, and redshift-space distortions. In this paper we assess several different methods for modeling the theoretical power spectrum, testing them against the Gigaparsec WiggleZ simulations (GiggleZ). We fit for a base set of six cosmological parameters, Omega(b)h(2), Omega(CDM)h(2); H-0, tau, A(s), n(s) , and five supplementary parameters n(run), r, w, Omega(k), Sigma m(v) . In combination with the cosmic microwave background, our results are consistent with the Lambda CDM concordance cosmology, with a measurement of the matter density of Omega(m) = 0.29 +/- 0.016 and amplitude of fluctuations sigma(8) = 0.825 +/- 0.017. Using WiggleZ data with cosmic microwave background and other distance and matter power spectra data, we find no evidence for any of the extension parameters being inconsistent with their Lambda CDM model values. The power spectra data and theoretical modeling tools are available for use as a module for CosmoMC, which we here make publicly available at http://smp.uq.edu.au/wigglez-data. We also release the data and random catalogs used to construct the baryon acoustic oscillation correlation function.