What kind of base do your bulbs have. The wedge base, 921 type LED bulbs, may need to be turned over to make then work. With the bayonet base bulbs the wiring may have to be changed. Since you have already switched the wires and it still didn't work, I would plug in the incandescent bulb to verify that there is still power to the socket. When you switched the wires, you switched only the wires to the bulb/socket that didn't work, right? If the light is the kind where the switch selects either 1 or both bulbs, maybe the switch is bad. The incandescent should show that.
No, you don't have to switch all your wires. The fact that one light works indicates that something else is wrong.