When supernova blast waves interact with nearby molecular clouds, they send slower shocks into these clouds. The resulting interaction regions provide excellent environments for the use of MHD shock models to constrain the physical and chemical conditions in these regions. The interaction of supernova remnants (SNRs) with molecular clouds gives rise to strong molecular emission in the far-IR and sub-mm wavelength regimes. The application of MHD shock models in the interpretation of this line emission can yield valuable information on the energetic and chemical impact of supernova remnants. New mapping observations with the APEX telescope in ^12^CO (3-2), (4-3), (6-5), (7-6) and ^13^CO (3-2) towards two regions in the supernova remnant W44 are presented. Integrated intensities are extracted on five different positions, corresponding to local maxima of CO emission. The integrated intensities are compared to the outputs of a grid of models, which combine an MHD shock code with a radiative transfer module based on the 'large velocity gradient' approximation. Results: All extracted spectra show ambient and line-of-sight components as well as blue- and red-shifted wings indicating the presence of shocked gas. Basing the shock model fits only on the highest-lying transitions that unambiguously trace the shock-heated gas, we find that the observed CO line emission is compatible with non-stationary shocks and a pre-shock density of 10^4^cm^-3^. The ages of the modelled shocks scatter between values of ~1000 and ~3000 years. The shock velocities in W44F are found to lie between 20 and 25km/s, while in W44E fast shocks (30-35km/s) as well as slower shocks (~20km/s) are compatible with the observed spectral line energy diagrams. The pre-shock magnetic field strength components perpendicular to the line-of-sight in both regions have values between 100 and 200uG. Our best-fitting models allow us to predict the full ladder of CO transitions, the shocked gas mass in one beam as well as the momentum- and energy injection.