Just as a note, paying off loans is not "spending."
Here's a few things to consider. First, you can't take lateness off your credit report. That can stay on there for seven years. However, it would also be on your report that something got paid and that will help. Your credit isn't going to improve quickly regardless, but it will start improving sooner if you do something about it.
But here's another funny thing. Part of what improves your credit is how many months straight you have made payments on time. If you have nothing to pay, you can't establish that record. In fact, one way of establishing credit is to get a credit card, use it for something every month, and then make sure it is paid every month. Or, a loan can work for this as well. People will intentionally take out loans when they have the money available simply for this purpose, especially people who don't like credit cards.
So there's something to be said about continuing to pay on one of those loans, assuming you can make the payments on time every month. But... you can also try to find a credit card (maybe one with a security deposit) and use that to try to establish credit again. But again, only as long as you can pay it on time every month. And good idea to just put something small on there, just so you have something to actually pay. If you don't think you can control yourself with a credit card, you don't want to go that route. But it sounds like the problem came from being out of a job. (Welcome to the economy. And by the way, congrats on getting a job.)
I would definitely pay off the 3K one. Sometimes it's not a bad thing to keep a student loan. They're usually pretty low interest, and this can be used to help get credit back. Not to mention that it's a good idea to have some emergency money set aside. And... I really don't know which would help your credit more -- just suddenly paying that entire loan off, or making payments on time and establishing a record.
As to the car... how badly do you need one? If you're in serious need of that car now, like if it's the only way you can get to work, then get it. If you can wait and say take a bus to work or whatever, then you may want to tuck some of that emergency money aside and then save with current earnings for a car later.
If you pay off both loans and buy the car, you've only got $2K left. That isn't very much. It's recommended to have 6 months worth of living expenses put aside.
There's different things you can do, and I think the circumstances of how badly you need the car, and what's going on with that student loan will make some differences. You can also pay say half that loan and make payments on the rest or something. You do want an emergency fund, but you also don't want to carry a lot of debt. If you pay part, then you have smaller debt, something you can get paid in a shorter time.
Here's what I'd probably do: pay off the $3K loan. Your credit report will show this debt as being paid off, and you'll be rid of some interest charges. Then set aside an emergency fund in case this job goes away. That would be my first two priorities.
Now, between the student loan, car and saving -- you need to consider stuff like what rate of interest on that loan, how long it would take to pay it off, what I said above about a credit card or loan helping to establish credit, and also how badly you need a car, and think seriously about the difference between need and want. And whether there's money left over after the emergency fund will make a difference. And really, just whether you want to be saddled with dealing with leaving one loan -- paying part of it is an option.
When you get the debt paid, it would help to get a credit card, charge some small thing each month, and pay it in full each month -- this will help to re-establish your credit. It will take some time. How late debt affects your credit isn't linear - it'll affect it more now and less later. You want to make the bad go away, but you also have to do something to establish good, and unfortunately in this world, that pretty much means credit cards and loans.
This is all just my opinion of course.
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