Cold fronts in cluster cool cores should be erased on short timescales by thermal conduction, unless protected by magnetic fields that are "draped" parallel to the front surfaces, suppressing conduction perpendicular to the sloshing fronts. We present a series of MHD simulations of cold front formation in the core of a galaxy cluster with anisotropic thermal conduction, exploring a parameter space of conduction strengths parallel and perpendicular to the field lines. Including conduction has a strong effect on the temperature distribution of the core and the appearance of the cold fronts. This results in the heating of this gas below the front on a timescale of a Gyr, but the sharpness of the density and temperature jumps may nevertheless be preserved. Finally, the combination of sloshing and anisotropic thermal conduction can result in a larger flux of heat to the core than either process in isolation.