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Hello, that is good trouble shooting work you have done so far.
The roundish pressure switch that the tube is connected to probably has a leaking diaphragm in it ...OR... the inducer blower wheel is clogged with soot and not moving enough air through the fire box (heat exchanger)...
Next step will be to use a U tube manometer to check the vacuum on the tube leading to the pressure switch. (the one you blew through)...to see if the vacuum exceeds the pressure rating stamped on the device. (approximately 0.02 to 0.04 inches water column).