Intentionally overwithholding tax in order to get a big refund is essentially loaning your money to the government for free. Since you have debts and other obligations, you're not in a great position to use the IRS as a free savings account. I would definitely adjust your withholdings so that you can use your money more effectively.
how many allowances will bring me to "net zero"?
You'd need to project out your total tax for the year, subtract out what you've paid year to date, and calculate how much needs to be withheld each paycheck from here on out. That's easier said than done, but the IRS has a calculator that may get you pretty close.
At worst, increase the allowances by 2 or 3, and if too much (or not enough) is taken out to get you close to your total tax from the year, then adjust it again. There's no harm in adjusting it multiple times throughout the year.
Or are there other opportunities I am missing?
What other expenses can you cut? Are you contributing to a 401(k)? You could temporarily stop this until you get your debts cleaned up. I'm assuming you have many years until retirement, so you have time to make up for the lost match.
Could you just sell the car and use the equity (if any) to get a cheap car for cash, if only for a year or two? I'm concerned that you're going to lower your payment by overpaying for a cheap used car.
Also, consider setting some aside for emergencies just so that you don't have to borrow money to cover unplanned expenses. Remember, though, as Dave says, Christmas is not an Emergency. You just need to plan for it. Set aside $50 a month and by Christmas you'll have $600 for gifts.
All in all, $2,500 in credit cards and one car payment is not a terrible position to be in. Obviously you make decent money if you're getting $5k in tax refunds, so it sounds like just a little discipline and sacrifice will get you where you want to be quicker than you think.
The 4 cards is about 5% of the total CC debt. And I'm not contributing to a 401k currently, and I'm older than you think.
So things are a little more bleak that originally posed. That means you have about $62K in CC debt. You haven't specified your income (which is fine), but it doesn't sound like changing your withholdings is going to fix the problem (it just changes when you have cash to pay off debt). Now if changing withholdings keeps you from adding more debt then it can have some benefit. But it's not going to move the needle.
You need to first get into a position to not add any more debt. One way to do that is to create a cash budget to make sure you don't spend more than you bring in. Then set small, achievable goals that you can accomplish in a short enough time that you can feel the progress. You can do other things to accelerate the process by selling things, getting side jobs, etc. Obviously you're familiar with Dave Ramsey so I can speak from experience that his plan can work if you are disciplined enough to follow it.