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Having your score pulled for a loan consideration does lower it. By single digit numbers.
You will be better served to pick your refi banks/credit unions carefully and refi.
Dropping your interest rate to single digits will save you a nice amount monthly of at least $50.00 a month.
Then you have easier solutions to your problem.
Unless you happen to be close to your EAS you should be able to get refinanceing/finance approval from NAVY FEDERAL CREDIT UNION.
Every eligible person should use Navy Federal as they are a great CU with outstanding customer service and great lending.
Do you have a NFCU account? If not go online right now and open one.
You should put your Direct deposits there as you get paid a day early.
Then right after you open a NFCU account you need to log onto USAA, and open an account, get an insurance quote. You will probably save about $50.00 a month in an apples to apples comparison going to USAA for your insurance. (And they have good banking to, just not as generoous as USAA- I do recommend opening a savings account through them though)
Then take a look at your deductable. Raising your insurance deductable from $250 to $500 an incident usually translates to about a $20.00 a month savings. Put the 20.00 a month away in a savings account, in a year you have your deductable increase put away in case you need it and at that point you are only saving money.
between NFCU refi, USAA for insurance, and raising your deductable you could easily lower your current monthly car commitment by between 100-150 a month or more. Cut whatever you save in half, pay half "extra" on you new car refi and keep the other as victory money.