I assume you're not a commercial hauler? CA is prohibiting commercial use of pre-emission control big rig diesels, but if you're not commercial the only issue you'll have towing a trailer into California in January will be possible restrictions over snowbound mountain passes.
If chain controls are in effect, you will be stopped unless you have chains for both the truck and at least one trailer axle. During a severe storm the roads may close entirely or be restricted to single vehicles until conditions improve.
Usually this is only an issue on the northern passes like I-5, I-80 and US 50. The southern routes like CA 58 over the Techahapi Mtns or the mountain passes north of LA ( I-15 over Cajon Pass or I-5 over the Grapevine) may occasionally have chain requirements but they seldom last longer than a day or two.
If you're moving to California, your F-250 will be fine as long as the engine hasn't been modified by removing or altering any of the factory supplied smog equipment. This includes tuner chips that don't have CARB approval. CA's smog laws take into account whether vehicles are manufactured to meet the CA or 49 state standards in effect when they were made and the allowed emission levels vary accordingly.
If your vehicle is in stock condition with intact factory supplied emissions equipment, you should have no trouble passing the required smog test to register the vehicle in CA. For diesels the test consists of a visual inspection of the emissions equipment and a test for excessive smoke.