This paper tries to establish whether there are variations of the central wavelengths or the profile shapes of diffuse interstellar bands (DIBs) and whether these variations are caused by different physical parameters of translucent clouds. For this purpose we used spectra of two stars seen through two different single clouds: HD34078 (AE Aur) & HD73882 acquired using two different instruments: the MIKE spectrograph, fed with the 6.5 m Magellan telescope at Las Campanas Observatory, and the UVES, fed with the 8 m Kueyen telescope at the Paranal observatory. The wavelength displacements of the DIBs at 6196, 6203, 6376, 6379 and 6614 AA with respect to the well known interstellar atomic and molecular lines (K{sc i} and CH) have been measured. The mentioned shift is seemingly absent in the DIBs at 4726, 4964, 4763, and 4780 AA. In addition the considered profiles may show (in HD34078) extended red wings. The observed phenomena are likely related to physical parameters of intervening clouds (rotational temperatures of molecular species) and may help in the identification of the DIB carriers.