Paying a lot more than appraised value seems like a bad financial idea imo.
However, it's your money, so *if* you have the money to do it...don't say we didn't warn you
Remember that when you go to sell, you are very unlikely to find buyers who feel the way you do about overpaying for this house. (I agree with stannius - by 'have the money' I mean you have the cash to make a much larger down payment without having to change your financial goals or "make up" that cash in the future.)
A few random ideas:
+Consider entering a written option with the potential sellers, that if they decide to sell you get right of first refusal? That wouldn't obligate you and could keep your dream alive while you get some time to process the $ and emotional impacts. Imo do this only if the sellers are willing to do it for a token fee.
+If you are on the fence about your decision and have the $, you could hire an appraiser yourself (any bank would still hire their own, but at least you'd have a better idea). Makes the most sense if appraisal fee is a small amount relative to home value.
+There may be other ways to pay/get value to the sellers than overpaying for the house - buying some of their personal property (art, furniture etc.) or giving them a big discount on something they need & you do (say you have a woodlot & they have a wood stove). Check with a lawyer & accountant before suggesting anything to the potential sellers. This is still a financial cost to you, but perhaps not quite as out there.