The cross-correlation function and template matching techniques have dominated the world of precision radial velocities for many years. Recently, a new technique, named line-by-line, has been developed as an outlier resistant way to efficiently extract radial velocity content from high resolution spectra. We apply this new method to archival HARPS and CARMENES datasets of the K2-18 system. After reprocessing the HARPS dataset with the line-by-line framework, we are able to replicate the findings of previous studies. Furthermore, by splitting the full wavelength range into sub-domains, we were able to identify a systematic chromatic correlation of the radial velocities in the reprocessed CARMENES dataset. After post-processing the radial velocities to remove this correlation, as well as rejecting some outlier nights, we robustly uncover the signal of both K2-18b and K2-18c, with masses that agree with those found from our analysis of the HARPS dataset. We then combine both the HARPS and CARMENES velocities to refine the parameters of both planets, notably resulting in a revised mass and period for K2-18c of 6.99+/-0.97 Earth masses and 9.2072+/-0.0065d, respectively. Our work thoroughly demonstrates the power of the line-by-line technique for the extraction of precision radial velocity information.