The only "face" you need to be thinking about right now is your car's, not yours. What's wrong with your face that makes you think that the sellers will quote you high prices anyway?
And you yourself explained how a friend of yours went from shop A to B to C and finally settled for a price from another shop. If he hadn't gone to all those shops, would he have known that 26,000 was the lowest he could ever get it for? You need to do the same.