MONOS.II. Orbit review and analysis for 35 single-lined spectroscopic binary systems and candidates

[ABRIDGED] AIMS: The MONOS project is collecting information and studying O-type spectroscopic binaries with delta > -20 deg. In this 2nd paper, we tackle the study of the 35 single-line spectroscopic binary (SB1) systems identified in the previous paper of the series (arXiv:1904.11385) by analyzing our data and reviewing the literature orbits of such systems. METHODS: We have measured the radial velocities for the ~700 spectra in our database using two different methods: Gaussian fitting for several diagnostic lines per object and cross-correlation using synthetic spectra. We also explored the TESS database and analyzed the light curves for 31 of the systems. RESULTS: We have confirmed 21 SB1 systems, discarded the binary nature of 6 stars (9 Sge, HD 192 281, HDE 229 232 AB, 68 Cyg, HD 108 and \alpha Cam), and left 6 stars as inconclusive due to lack of data. The remaining two stars are 15 Mon Aa which has been classified as SB2, and Cyg OB2-22 C, for which we find evidence that it is most likely a triple system where the O star is orbiting an eclipsing SB1. We have also recalculated 20 new orbital solutions, including the first spectroscopic orbital solution for V747 Cep. For Cyg OB2-22 C we have obtained new ephemerides but no new orbit.