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The hits will affect you in the short term but not so much after a few months or a year. From everything I've read Walmart and Amazon are easy to get as long as you have a score, but Discover is much harder. Let your credit sit for a few months and then see where you are. If you don't already have one find a good credit union and become a member. My scores are probably lower than yours and I just got approved for a car loan at 10.29% through my credit union, they flat out told me if I wasn't already a member I wouldn't have gotten it. A good history there could really help when it comes time for the mortgage. And possibly refinancing your new car loan.
The car loan will lower your scores in the short term as well but as you make a few payments it will get better and better.