Cheri -
I'm not sure which running lights you are referring to - the lights on the body of the camper, or the so-called parking lights in the taillight assemblies. Either way, loss of a group of lights often indicates that a ground connection has gone bad. The grounds are always in or near the light socket. If you open up the light assemblies (pull the lens off), you should see a ground connection from one or more of the brass sockets that hold the bulbs. Wiggle those connections, make sure they are tight.
A more definitive check if you are equipped to make it is to turn on the lights, remove the bulb that isn't lighting up, and use a voltmeter to check the voltage from the center contact of the socket to a known good ground. If 12 volts appears at the socket, but the bulb won't light, then the problem is definitely a bad ground.
By any chance, do you have a copy of the wiring diagram that came in the original owner's manual for your TM? If so, that would help a lot.
Bill
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