I'm new here, but I recently attended a seminar sponsored by a government housing agency. The subject was cleaning up your credit so that you can buy a home.
You have an auto loan. It is your only debt. It is impacting your FICO score in two ways: (a) first, the credit ratio (as a percentage of credit utilized to credit limits approved) and (b) payment history. You didn't mention what kind of FICO score you have now or what your income level is in comparison to the loan. Let's say you have a credit card with $1000 limit and you only charge $100 a month and pay it off; you have a credit ratio of only 10%; on the other hand, if you charge $500 a month or carry a revolving balance of up to $500 a month, then you have a credit ratio of 50%. The higher the credit ratio, the lower the FICO score. As has been said better elsewhere on this forum, FICO scoring likes to see low utilization of credit limits -- ideally, under 30%. Installment loans are a little different than revolving credit, but here's where you get in trouble. If you have no revolving credit accounts and the only credit utilization is the installment car loan, then you are going to have a high credit ratio. Fortunately, FICO understands that installment loans ALL begin with 100% credit utilization, and so the emphasis in the FICO matrix is to see incremental decreases in the balance (which you say you are doing). Another way that the car loan benefits you is that other lenders want to see that others are giving you credit -- however, they want to see more than a 3-month payment history.
You should make your payments every month on time. Then, after paying your bills and putting away money for your emergency fund, you can put a little extra towards the car payment (add $50 to each payment or make another payment two weeks after the regular payment). The important thing is to not miss any payments or be late on any payments. And you can't pay in advance -- that is, you can't pay two months' worth of payments in one month and then skip the next month. You have to make the regular payments each and every month before the due date.
I was told that when it came to installment credit, the MOST IMPORTANT thing was to make timely payments over an extended period of time. The balance amount is not nearly as important as the length of the installment account and how long you have been making timely payments. The justification I was given is that an installment account (like a car loan) tells a lender much more about how you will pay a mortgage (as opposed to a revolving credit account). Paying off a car loan 3-5 years early will mean that you only have 2 years of payment history and, depending upon other credit factors, may do more harm than good -- not to mention that instead of prepaying the car, you could be saving that money for a downpayment on a house so that you can then take advantage of the mortgage interest tax deduction which will save you thousands on your taxes. And a good mortgage rate begins with a stable credit history and a nice downpayment.
As for an educational loan -- it's treated the same as an installment loan, but we were told that it really isn't given that much weight (unless you just graduated from law school and owe $100,000). The reasoning is that "lenders assume that school loans were granted pretty much regardless of credit worthiness." This changes, of course, if you are late on your payments or - God forbid - default on the loan. Then the loan becomes extremely detrimental to your FICO score.
That's what we were told anyway. IMHO, you are going to need at least a couple of years of regular monthly payments, so you should make your car payment as you agreed and then put your extra cash into savings while you have the cash flow to do so.