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I'm not sure what the "sunshine" route is, but I can say that there is no cross country that can guarentee good weather in winter. Even if you took I-10 to I-5, you'd have the potential for ice and snow From Texas through Arizona, and then would be dealing with mountain passes that also see snow along I-5.
Direct routes are usually better bets, simply because it means you'll be on the road for the least amount of time. However, when you are about to depart, you can use current forecasts to help route around storms if they are in your path.