I work at a dealership. Some dealers wont move regardless of payment type. The words Im paying cash is worse for us not better. In fact saying that instantly makes us want to raise the price not lower as we will get paid no backend money.
My dealership is a 1 price store like carmax. You can work us till your blue in the face and pennies will not have changed sides. We do price ours "apples to apples" to be the lowest price. Some will still try the old I saw it lower somewhere else, if thats true show me the ad and Ill beat it if its apples to apples. It never is.
As for paying off, in Florida you can pay off anytime you want you only pay interest on the time you held the loan. So if its 15 days then you pay 15 days which is dollars if anything. Again here in Florida there are also no prepayment penalities. Some places will have a penalty if you pay off early.
If you finance with the dealer you wouldnt return to the dealer to pay if off, you would be paying the finance company not the dealership.