I think the biggest mistake(s) you made were drawing cards with your Governors early on. You draw Turn 7, when your opponent has 6+ a pearl diver in hand. You buy a Gold with your extra cards, and allow him to buy a Province. You could have just gained the Gold instead.
Then on Turn 9 you draw again, buying a province, but you also increase his hand from 7 to Province.
You draw again on turn 10, which lets you send back 2 estates, but then he buys a Province again (he had to Governor draw to get it though, this one not so bad).
So... Governor is really fast against Ambassador if you keep drawing cards for your opponent!