New results are presented on a high-statistics measurement of Collins and Sivers asymmetries of charged hadrons produced in deep inelastic scattering of muons on a transversely polarized ^{6}LiD target. The data were taken in 2022 with the COMPASS spectrometer using the 160 GeV muon beam at CERN, statistically balancing the existing data on transversely polarized proton targets. The first results from about two-thirds of the new data have total uncertainties smaller by up to a factor of three compared to the previous deuteron measurements. Using all the COMPASS proton and deuteron results, both the transversity and the Sivers distribution functions of the u and d quark, as well as the tensor charge in the measured x range are extracted. In particular, the accuracy of the d quark results is significantly improved.