Just a touch more info needed. When you swapped the two wires, I assume you had to move the switch to the 'Reverse' position to get it to go forward, albeit with a push start. If that's the case, I don't think there is anything wrong with your motor. The same coils are used to start it in either direction - just reverse polarity used on the start coil for reversing.
That would indicate that there is something wrong with the drum switch. It's not feeding power to the starting coil in the 'Reverse' position. Unless, of course, you are using a wrong wire in there someplace. Should probably be the #5 and #8 wires for the direction selection.