Yes, you should definitely shop around, but I would also offer another suggestion. Presumably you want this to be your car, and someone else is helping with the loan. However, you both have to be on the title and on the loan.
Since you are both going to be on the title and loan anyway, you might consider switching this around. Have your cosigner go first, and you cosign the loan. That way, they look at this as loaning to a 750 score first.
It might also help if you can find a credit union you can join. Around here they always use the high credit to establish the rate. In your case, the rate would be based on the 750 score, regardless of what your score is.
Just for the record, 5.7% for 72 months is not unreasonable right now for bank financing.
You do not mention what kind of car you are buying or whether it is new or used. Since you have said you are going 72 months, presumably the car can't be all that old. Unless you are already set on one specific vehicle, it would not hurt to shop for the current deals being offered. There are new cars being offered at 0% for 60 months. Since you are looking at a used car at 5.7% for 72 months, you may find that the payment is not that much different. Note that a loan on $18,000 for 60 months at 0% is $300 a month, and $18,000 at 5.7% for 72 months is $296 a month.
It is possible that a manufacturer offering 0% may approve a deal with you on it just to help sell a car. In that case, it's a yes or no decision, not a question of how much the rate should be. It would certainly be worth a try if a new car makes sense to you.
If you are looking for used, some manufacturers offer lower interest rates for certified cars as well.
If you are not taking a special rate offer from a manufacturer, one thing you should watch out for is that most dealers will try to jack up your rate just to make more money on your deal. They will use the excuse that your credit is making them raise the rate. That is where the shopping aorund comes in. You should know what you can get before you get stuck with what the dealer is offering.