Massive stars are a key element to understand the chemical and dynamical evolution of galaxies. Stellar evolution is conditioned by many factors: rotation, mass loss, and interaction with other objects are the most important ones for massive stars. During the first evolutionary stages of stars with initial masses (i.e. M_ZAMS_) in the M_ZAMS_~18-70M_{sum}_ range they are of spectral type O. Given that stars in this mass range spend roughly 90% of their lifetime as O-type stars, establishing the multiplicity frequency and binary properties of O-type stars is crucial for many fields of modern astrophysics. The aim of the MONOS project is to collect information to study Northern Galactic O-type spectroscopic binaries. In this second paper, we tackle the study of the 35 single line spectroscopic binary (SB1) systems identified in the previous paper of the series Maiz Apellaniz et al., (2019, Cat. J/A+A/626/A20) analyze our data, and review the literature on the orbits of the systems. We have measured ~4500 radial velocities for a selection of diagnostic lines for the ~700 spectra of the studied systems in our database, for which we have used two different methods: Gaussian fit for several lines per object and cross-correlation with synthetic spectra computed with the FASTWIND stellar atmospheric code. We have also explored the photometric data delivered by the TESS mission to analyze the light-curve (LC) of the systems extracting 31 of them. We have explored the possible periods with the Lomb-Scargle method and, whenever possible, calculated the orbital solutions using the SBOP&GBART codes. For those systems in which an improved solution was possible we have merged our RVs with those in the literature and calculated a combined solution. As a result of this work, of the 35 SB1 systems identified in MONOS I, we have confirmed 21 systems as SB1 with good orbits, discarded the binary nature of 6 stars (9 Sge, HD 192281, HDE 229232 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.