"Is it worth driving to either state to play a large number of lottery tickets with those odds for the sake of remaining anonymous?"
Given the odds of wining there's very little sense in buying tickets at all, but unless you're going to by so many tickets that the cost of driving becomes negligible it makes no sense at all to drive far out of your way.
As a raw statistic the chance of dying in a traffic accident is 1 in 75 million for every mile you drive. Those aren't bad odds at all when they're the odds of dying, but it means that if you drive a mile to buy one ticket you're 4 times as likely to die as you are to win the jackpot. If you drive 50 miles to another state and buy 50 tickets you'll actually be 8 times as likely to die as to win because you'll also have to drive back home. The IRS figures that 100 mile r/t will cost you about $58 in vehicle expenses, so if you buy 58 tickets you've spent $3 per ticket instead of $2.
If you bought your ticket at a place you were driving past near home you could spend the same $3 to buy 50% more tickets and have a better chance of winning any prize, without the slightly higher chance of dying in a traffic accident. If you're not going to buy 50 or 58 tickets the increased risk and cost and risk per ticket are both higher.
Then there's the tax rates in the different states. If the other state has a higher income tax you'll keep less of any large prize. If you win anything that gets reported you'll be filing a tax return in the other state, and that potentially includes paying $100 or more for paperwork resulting from a fairly modest prize. You'll also need to trust the mail or make another trip to collect any prize you win, though that would eliminate the added cost of buying tickets there that time around.
And, of course, even if you buy thousands of tickets it's till very unlikely that you'll win a prize that's big enough to matter.
That said, if you've got another, sensible, reason to be in a different state it may be a good idea to buy your ticket while you're there. If I were to win a jackpot with a NY ticket I'd pay the new 10.3% rate on taxable income between $5 and $25 million and 10.9% on anything over $25 million. I'm out of state several times a year, so if one of the lotteries is high enough I'll get tickets for the next 8 to 10 drawings while in one of the states with a lower tax rate. I'd have to move before claiming the prize, but I'd be okay with that. If your state's tax rate is lower you'd have to decide if it's worth paying the higher taxes to hide your name without creating a trust or LLC. Of course it should be noted that there are good reasons to have a trust even if your net worth is fairly modest.