Is your "big truck" a 4WD pickup? Are you accustomed to driving it in winter conditions?
Ten to twelve days using a route that (mostly) avoids snow and mountains isn't very much for a transcontinental drive in January. Remember too that daylight hours will be very short, so the length of your driving days is going to be curtailed.
The safest way to avoid (most) of the potential snow/ice conditions would be to head as far south as fast as you can, then follow the southern tier across the country to California, then up the coast through Oregon and to Seattle. Here's a map showing the "ideal" route for this -
https://goo.gl/maps/9YcWfdZBicN2 . You'll see that Google Maps estimates 70 hours for this trip, but honestly Google tends to be very optimistic in this regard, so probably 80 or 90 is more like it. Those are "wheels turning" hours, not counting fuel, breaks, dog walks, food, etc.
Now a more direct route, e.g.
https://goo.gl/maps/f6D1UptS8vk - will take far fewer driving hours, you'll run the risk that weather might slow you down, or even stop you, for a day or more en route, and you'll probably be in winter driving conditions the whole way. Now if you have good winter tires and stick to the interstates during daylight hours, you're probably going to be okay, but the going will be slow. It can be bitterly cold (well below zero F) in places like the Dakotas or eastern Montana at that time of year, along with strong winds and wind chill. It's just a fact of life. There are some mountain passes - the continental divide in Montana, Lookout Pass between Missoula and Idaho, and Snoqualmie Pass east of Seattle, where you might get stopped for a day, or where you'll have to "chain up." No party, but thousands of people do these trips every year and almost nobody gets eaten by wolves, so you'll manage.
However, if it was me in your circumstances I'd look at two options. First, look at shipping your vehicle while you and your family and pooch fly. Transcontinental auto shipping will probably cost around a grand, but it would take the fear factor away. (Not knowing your circumstances, I'd also say you could investigate selling the truck and buying another vehicle on the west coast, but that might be a bit extreme.)
Second, just make more days available, do the southern route, and enjoy yourselves. You could stop, for example, at Joshua Tree National Park in southern California, go to some Oscar candidate movies in Hollywood. Or stop at Hearst Castle and see the elephant seal breeding colony on the beach just up the road from the castle parking area. Visit the Monterey Bay area in California, and see the migrating monarch butterflies (million of them) in the various refuges along the coast. Travel through the redwoods, awesome in the mist at that time of year, and maybe do some storm watching on the spectacular Oregon coast. But you'd need more than 10 or 12 days for that.
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