We present an analysis pipeline that enables the recovery of reliable information for all four Stokes parameters with high accuracy. Its novelty relies on the effective treatment of the instrumental effects even before the computation of the Stokes parameters, contrary to conventionally used methods such as that based on the Mueller matrix. For instance, instrumental linear polarization is corrected across the whole telescope beam and significant Stokes Q and U can be recovered even when the recorded signals are severely corrupted by instrumental effects. The accuracy we reach in terms of polarization degree is of the order of 0.1-0.2%. The polarization angles are determined with an accuracy of almost 1{def}. The presented methodology was applied to recover the linear and circular polarization of around 150 active galactic nuclei, which were monitored between July 2010 and April 2016 with the Effelsberg 100-m telescope at 4.85GHz and 8.35GHz with a median cadence of 1.2 months. The polarized emission of the Moon was used to calibrate the polarization angle measurements. Our analysis showed a small system-induced rotation of about 1{deg} at both observing frequencies. Over the examined period, five sources have significant and stable linear polarization; three sources remain constantly linearly unpolarized; and a total of 11 sources have stable circular polarization degree m_c_, four of them with non-zero m_c_. We also identify eight sources that maintain a stable polarization angle. All this is provided to the community for future polarization observations reference. We finally show that our analysis method is conceptually different from those traditionally used and performs better than the Mueller matrix method. Although it has been developed for a system equipped with circularly polarized feeds, it can easily be generalized to systems with linearly polarized feeds as well.