I just figured out why the Newport 6000 isn't stabilizing the temperature. It is designed to drive a TEC, so that when the temperature is too high, it just applies a negative current. Of course, this doesn't work with a resistive heater; it just keeps heating it up more.

I'm not sure if Frank has actually used this with a restive heater before, but it doesn't appear that you can limit the low-current level or add an offset.