The high accretion rates needed to fuel the central black hole in a galaxy can be achieved via viscous torques in thick disks and rings, which can be resolved by millimeter interferometry within the inner ∼20 pc of the active galaxy NGC 1068 at comparable scales and sensitivity to single dish observations of the Circumnuclear Disk (CND) in the Galactic Center. To interpret observations of these regions and determine the physical properties of their gas distribution, we present a modeling effort that includes the following: (i) simple dynamical simulations involving partially inelastic collisions between disk gas clouds; (ii) an analytical model of a turbulent clumpy gas disk calibrated by the dynamical model and observations; (iii) local turbulent and cosmic ray gas heating and cooling via H₂O, H₂, and CO emission; and (iv) determination of the molecular abundances. We also consider photodissociation regions (PDRs) where gas is directly illuminated by the central engine. We compare the resulting model datacubes of the CO, HCN, HCO⁺, and CS brightness temperatures to available observations. In both cases the kinematics can be explained by one or two clouds colliding with a preexisting ring, in a prograde sense for the CND and retrograde for NGC 1068. And, with only dense disk clouds, the line fluxes can be reproduced to within a factor of about two. To avoid self-absorption of the intercloud medium, turbulent heating at the largest scales, comparable to the disk height, has to be decreased by a factor of 50-200. Our models indicate that turbulent mechanical energy input is the dominant gas-heating mechanism within the thick gas disks. Turbulence is maintained by the gain of potential energy via radial gas accretion, which is itself enhanced by the collision of the infalling cloud. In NGC 1068, we cannot exclude that intercloud gas significantly contributes to the molecular line emission. In this object, while the bulk of the X-ray radiation of the active galactic nucleus is absorbed in a layer of Compton-thick gas inside the dust sublimation radius, the optical and UV radiation may enhance the molecular line emission from photodissociation regions by ∼50% at the inner edge of the gas ring. Infrared pumping may also increase the HCN(3−2) line flux throughout the gas ring by about a factor of two. Our models support the scenario of infalling gas clouds onto preexisting gas rings in galactic centers, and it is viable and consistent with available observations of the CND in the Galactic Center and the dense gas distribution within the inner 20 pc of NGC 1068.