The "sloshing" of the cold gas in the cores of relaxed clusters of galaxies is a widespread phenomenon evidenced by the presence of spiral- shaped "cold fronts" in X-ray observations of these systems. In simulations, these flows of cold gas readily form via interactions of the cluster core with small subclusters, due to a separation of the cold gas from the dark matter (DM), due to their markedly different collisionalities. In this work, we use numerical simulations to investigate the effects of increasing the DM collisionality on sloshing cold fronts in a cool-core cluster. For clusters in isolation, the formation of a flat DM core via self-interactions results in modest adiabatic expansion and cooling of the core gas. In merger simulations, cold fronts form in the same manner as in previous simulations, but the flattened potential in the core region enables the gas to expand to larger radii in the initial stages. Upon infall, the subclusters DM mass decreases via collisions, reducing its influence on the core. Thus, the sloshing gas moves slower, inhibiting the growth of fluid instabilities relative to simulations where the DM cross section is zero. This also inhibits turbulent mixing and the increase in entropy that would otherwise result. For values of the cross section σ/m ≳ 1, subclusters do not survive as self-gravitating structures for more than two core passages. Additionally, separations between the peaks in the X-ray emissivity and thermal Sunyaev-Zeldovich effect signals during sloshing may place constraints on DM self-interactions.