I'm not an expert, and am not familiar with the devices you are using, but that circuit seems to be not quite suitable for using an FET directly in place of the Darlington. The best thing to do might be to try it first with a proper Darlington, or a couple of BJTs in a Darlington configuration.
FETs must be switched fully, and quickly, to run cool. Also, many of them are typically used with a small resistance in series with the gate, which might help to prevent some ringing/oscillation that might be contributing to your overheating problem.
Depending on the driving voltage and the FET specs, you might have to add a small BJTransistor (or other) switching circuit, to use part of the power supply rail to drive the gate, just to drive the gate hard-enough for the FET to work well-enough, if you insist on using an FET.
It can be made to work. Maybe you should google for FET OR MOSFET switch OR switching circuit OR circuits.
- Tom Gootee
http://www.fullnet.com/~tomg/index.html