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So far it sounds like there might be a crack or hole in the heat exchanger, and when the blower comes on it blows air into the fire box and blows the flames away from the flame sensor.
Watch the burner flames carefully, notice the shape of the flames, how they look, the look for changes when the blower starts... if you see any change, the heat exchanger is bad.
Let me know what you find, we can go from there with no time limit.