We present the results of a new high-resolution study of the molecular gas associated with the supernova remnants (SNRs) G349.7+0.2 and G18.8+0.3. The observations were performed with the SEST telescope in the 12 CO J = 1-0, 2-1 and 3-2 lines (beams of 45, 23 and 15, respectively). The present observations have provided, for the two SNRs, new evidence in support of the existence of physical interaction between the SN shocks and the adjoining molecular clouds. In the case of G349.7+0.2, the new observations revealed for the first time the internal structure of the shocked cloud, as well as the kinematical consequences of the impact of the SNR shock on the molecular cloud. From these observations we were able to constrain the conditions of the pre-shocked gas. The molecular cloud associated with G349.7+0.2, centered near ν LSR = +16.2 km s -1 , has a linear size of about 7 pc, a mass of ∼10 4 M ○. and a volume density of ∼10 3 cm -3 . The high line ratios derived are indicative of the existence of shocks in the cloud. From the asymmetries observed in the line shapes we propose that the SN shock cloud is running into the denser part of the cloud and has probably begun to disrupt it, pushing the eastern component clumps away from us, and the western fragments toward us. After comparing our estimates of the column density of the intervening gas with similar calculations based on ASCA X-rays spectral fitting we conclude that the best way to make these results compatible is by assuming that the associated cloud is placed behind G349.7+0.2 along the line of sight, and the SNR/molecular cloud encounter is taking place on the far side of the SNR. This model also provides a natural explanation for the lack of strong X-ray absorption in the central region of G349.7+0.2. Evaporation of part of the associated cloud must be responsible for the central X-ray emission. The comparison with IRAS infrared data provides additional support for the hypothesis of SNR/cloud physical interaction. From the study of the molecular gas in the neighborhood of the five OH (1720 MHz) masers detected in G349.7+0.2 we find that in three cases the maser peak velocity coincides with the local CO peak velocity, while in the remaining two cases the maser peak velocity agrees with a secondary, blended CO component. We conclude that the masers are excited at the sites where a non-dissociative C-type shock, locally transverse to the line of sight (or forming a large angle with it), hits a denser molecular clump. For the SNR G18.8+0.3, the new higher resolution observations have revealed excellent morphological agreement between one of the cloud components and the SNR shock front towards the eastern limb. The associated molecular mass is estimated to be ∼4.4 x 10 4 M ○. and the cloud volume density ∼1200 cm -3 . The analysis of the line ratios in this case revealed a maximum of R 2-1/1-0 = 1.25 at a position that exactly matches an indentation in the radio continuum emission in the remnant's shell, providing additional evidence of SNR/molecular cloud interaction.