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Your answer is simple. If you are sure you owe the collections are they are verified, send Pay-For-Deletion letters for the collections. Otherwise you can dispute them as not yours or inaccurate and possibly having them deleted. Deletion is the ideal outcome. Pay down credit card to less than 10%. Since those are your only baddies you are very close to having zero baddies!!
Having an installment loan for a car will certainly help your credit file.
...your best bet is to join a local CU and start making regular deposits in a savings account there in the amount you anticipate your future car payments will be ...make those regular deposits your first priority and don't touch the money ...you're showing the CU that you can make regular payments so that when you ask them for an auto loan (after your credit is fixed) they will be more inclined to do so ...would also put that tax return in the CU savings when you get it.
...next, get your CC utilization down to under 10% and keep it there ...you can spike it temporarily if necessary but pay it down again before the statement date.
...then start on your CA accounts ...assuming they are valid your only leverage is offering to PIF for deletion from the CRAs ...many CAs are hard-nosed and won't do PFDs but money talks and most will ...your worst case is that you PIF and the CRAs reflect that ...still much better than unpaid collection accounts.
...once your CRA records are "fixed" to the extent possible and you have a stable CU deposit history, go talk to the CU loan officer about a car loan ...if necessary you can offer your savings account as security ...you can always get a loan from a dealer but with your credit history you'll pay a usurious interest on it, your CU will give you a MUCH better deal on a loan than anywhere else and you'll have taken a major step in building your credit history.
...and once you've made several loan payments as due, it will be time to go back to the CU loan officer and ask for a CC with a decent credit limit ...from that point on you're on the road to a full credit recovery ...hth