Can the dealer change the price after we agree?

A written, signed contract generally locks in the price, but a verbal agreement or an unsigned quote may not. Get the agreed out-the-door price in writing and compare every line of the final contract against it before you sign. Be cautious of "spot delivery" situations where you drive off before financing is final and are later asked to sign new terms. The CFPB (consumerfinance.gov) and FTC have consumer guidance on auto financing if something changes unexpectedly.