With a 401(k) plan, CA requires withholding, but you can waive it. From the TSP site, it appears that they do not withhold state income taxes at all. That being said, you should check with your tax preparer or see if CA's Dept. of Revenue has a calculator. You can figure it out this way:
Regular gross income + cash distribution from TSP + other taxable income = total gross for the year. Plug that into a calculator to see where you stand in CA.
20% mandatory federal withholding applies if you take a lump-sum cash payout--there's no way around it.
Another downside to taking it as cash is you lose the tax-deferred status and it's added to your income, PLUS subject to a 10% penalty if you're under age 59 1/2. One of the most under-discussed disadvantages is the opportunity lost by not leaving it invested for retirement. You'd be surprised how much a small amount can grow between now and retirement if you leave it invested for retirement.
I'm not sure if the investment options are the same in the civilian TSP and the military TSP, but know the military TSP has a very limited scope of investments.
I would concur that the best thing for you to do is open an IRA. You can open one at most places with as little as $500 or $1,000. If you don't owe that much on your car, paying it off now at the loss of the investment returns and higher taxes this year is very short-sighted. Over the long-run, an IRA is much, much better.