The COMPASS Collaboration performed measurements of the Drell-Yan process in 2015 and 2018 using a 190 GeV/c π^{-} beam impinging on a transversely polarized ammonia target. Combining the data of both years, we present final results on the amplitudes of five azimuthal modulations, which correspond to transverse-spin-dependent azimuthal asymmetries (TSAs) in the dimuon production cross section. Three of them probe the nucleon leading-twist Sivers, transversity, and pretzelosity transverse-momentum dependent (TMD) parton distribution functions (PDFs). The other two are induced by subleading effects. These TSAs provide unique new inputs for the study of the nucleon TMD PDFs and their universality properties. In particular, the Sivers TSA observed in this measurement is consistent with the fundamental QCD prediction of a sign change of naive time-reversal-odd TMD PDFs when comparing the Drell-Yan process with deep inelastic scattering. Also, within the context of model predictions, the observed transversity TSA is consistent with the expectation of a sign change for the Boer-Mulders function.