A theoretical investigation has been made on linear and nonlinear propagation of the dust-ion-acoustic and dust-acoustic waves in a dusty plasma with vortex-like plasma particles. The linear dispersion relation for DIA and DA waves in an unmagnetized dusty plasma with vortex-like electron distribution and negatively charged dust particles have been analyzed elaborately. The basic characteristics of such waves have been theoretically investigated by using the reductive perturbation technique. The effects of external magnetic field, vortex-like negative ions, and arbitrarily charged static dust on DIA SWs and their multi-dimensional instabilities have also been studied. It has been found that these effects not only modify the basic properties of the DIA SWs, but also introduce some important new features. The basic features and the underlying physics of the DIA solitary structures have been also explicitly identified. It has been also found that the instability criterion and its growth rate are significantly modified by the external magnetic field and vortex-like distribution of negative ions. The implications of our results in space and laboratory plasmas are briefly discussed.