We present new experimental results on the dielectric constant in orientational glasses K$_{1-x}$Li$_x$TaO$_3$ (KLT) with x < 0.05, together with the detailed (analytic and numerical) study of a model which attributes the observed aging to the motion of the wall of polarization domains. We show that the dielectric constant after positive temperature jumps goes through a maximum as a function of the subsequent time. This observation and those previously reported (aging, cooling rate dependences, etc.) are compared with the predictions of the model, in which the variations of the dielectric constant are attributed to the change of polarization domain wall area. The total area decreases by domain growth and increases by nucleation of new small domains inside the larger ones. These two opposite variations are both hindered by static random fields (equivalent to energy barriers) due to the frozen dipoles borne by the off-center Li+ ions. Many results are well explained by a model with a single energy barrier. However, some effects can only be understood if a broad distribution of energy barriers is assumed. We use the experimental data to determine this distribution and find it to be unimodal with a width comparable to its most probable value.