93 Ford Taurus will not idle cold

93 Ford Taurus, 63000 miles, auto trans. When cold starting you must accelerate until the temp gets to normal or the engine will die. After reaching normal temp it will idle fine. When driving a short distance the idle gets rough but it does not die when stopped. When the idle got rough again I noticed the temp gauge that gone back to cold. Any help on what to check or replace will be appreciated.