We have investigated the effects of relaxing the normal assumption of frozen in convection on studies of radial instabilities in 0.6M⊙ carbon-oxygen white dwarfs with either pure hydrogen layers overlying pure helium layers or 0.6M⊙ carbon-oxygen white dwarfs with pure helium surface layers. In this paper we assume that convection can adjust to the pulsation at a rate determined by the time scale of a convective eddy. Using this assumption in our analysis stabilizes most of the modes in both the DA and DB radial instability strips. We also find that the blue edge of the DA radial instability strip, assuming frozen in convection, is between 12,0O0K and 13,000K. The blue edge for the DB radial instability strip (frozen in convection) is between 32,000K and 33,000K.