We develop a stochastic model for the dynamics of a dense fluid of flexible linear macromolecules. A polymer is represented by a harmonic chain of beads whose mobilities fluctuate in time between two values. A bead in the low-mobility state does not execute local motions, but may move by a cooperative slithering process involving the entire chain. A bead in the high-mobility state may execute both local and slithering motions. The rate at which the mobilities fluctuate is determined self-consistently as a function of chain length through an ansatz that associates these fluctuations with the configurational relaxation of neighboring molecules. We calculate the viscoelastic shear modulus and the coefficients of shear viscosity and self-diffusion for this model. The coefficient of shear viscosity eta shows three regimes of dependence on chain length N. For a fluid of short chains, eta similar to N, in agreement with the Rouse model and with the behavior of laboratory polymers. For a liquid of longer chains, eta displays an N dependence that is intermediate between N-3 and N-4, in agreement with laboratory measurements. In the asymptotic limit of large N, eta similar to N-3, in agreement with the prediction of the tube model.