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Perhaps the biggest myth of winter travel is that you should or need to "go south." The reality is that you can't go far enough south in the US to avoid the chance of seeing snow and ice. The only thing that will tell you where you can avoid bad weather is to look at a short term forccast just before you leave. It's a much better bet to plan to take the shortest route, which puts you on the road for the least amount of time, and gives you the most options if you do see bad weather.

As far as your actual plan, you're already pushing the limits of how far you can get in 2 weeks. You're looking at a minimum of 3 full days to get to Phoenix, and then one more to get to San Diego. Factor in an extra day or two just in case you do see some bad weather, and that only leaves you a day or two each in the 3 cities you want to visit.