The basic premise is a major flaw: the only things that could shift the Earth's orbit significantly are a near passage of or a collision with an astronomical body of sufficient size to alter the course of a body weighing 6.6 sextillion tons - the mass of the Earth. Either event would cause sufficient disturbances that there would mostly likely be no cities left standing, nor any people left in them to debate staying or fleeing - ergo, no story. (Of course, this is the Twilight Zone, where physics are not the same as in our world.)
The heating scenario seemed to imply extended or constant daylight, while the cooling scenario seemed to imply extended or constant darkness. In reality, the Earth's rotation would continue, and the night side of a heating Earth would probably undergo catastrophic weather as warm air was displaced to the night side; the day side of a cooling Earth would be a brief respite to the frigidity that is increasing. Admittedly, whatever catastrophe changed the orbit may have changed the rotation speed, but probably not arrested rotation to the point equaling "tidal lock".
A global disaster such as that portrayed in this episode would have caused the government to declare an emergency. At the time of production of this episode people would have been instructed to tune their radios to 640 or 1240 Kc (Kilocycles) for the Conelrad (Control of Electromagnetic Radiation) bulletins. But when we hear the announcer on Norma's radio it is shown to be tuned to exactly 700 Kc.