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Cutting to the chase, you're worried about all the wrong thing.

Of course it's feasible to drive from Houston to Portland, even in winter, but not with the approach you've laid out.

It's 2300 miles on the shortest route - doing that without stops and/or just taking a nap at rest area is in a word, homicidal. A trip of this distance requires a bare minimum of 4 full days on the road, and that's in good conditions. If you see snow or ice - something that's relatively likely on a trip of this size in January - then it's you'd need to factor even more time. "Winter gear" isn't nearly as important as experience when driving in winter conditions, and as you've stated, you don't have any experience, so that adds to the challenges and the need for extra time.

If you attempt to make this drive without proper rest, then the odds of you getting into a crash or going off the road and getting stranded in the middle of nowhere or in a snowbank are really quite high. Getting proper rest while sitting inside a car, that's essentially provides no insulation, when the temperatures are well below freezing is virtually impossible. In other words, doing this to "save money" is pennywise and pound foolish.

Driving yourself will probably be less expensive, and if you enjoy driving as you say, it could really be a worthwhile and enjoyable trip. But from the start, you need to have at least 5 days available (factoring at least 1 extra day for possible weather delays), and money in your budget to find actual places to sleep at night, so you can be rested enough to continue safely operating a 2 ton machine at 70+ mph. If you can't afford that, then the $1k to ship your car and fly there is really your only option.