Running tail light problem

Sounds like the sockets are corroded or the wiring to them is, actually common for this to happen over time. Check the socket first, if it's corroded, replace it. Another test if it's out. Give the light housing a good whack with your hand while it is still installed. If the light comes back on you have a bad bulb or socket. If it doesn't then pull the light out, take the brown wire with white stripe and move it around and see if the light goes out in one spot, also pull on it in both directions to see if it is a bad connection or the wire is corroded inside the insulation.