First shot at it would say you have a wire that has rubbed through and is shorting outor a loose connection, (although a loose connection wouldn't run your battery down.). If this is the case at some point in time it will make a permanent short and you will have no running lights.
Hopefully your running lights are fused separate from your ouher lightd which will maake it easier to trace and also allow you to get where you can fix it.
Are you referring to your parking lights, taillights, and clearance lights? All the lights that come on with the first click of your headlight switch? And are they blinking sporadically or rhythmically like a turn signal?
We will hope that after this long you have found the problem, but just in case I'll offer a couple of thoughts. Do they blink in a regular pattern as though they were running from a flasher? If so you should be able to find the problem circuit by listening for the flasher. Then try pulling the fuse to that circuit. If it stops, the problem could be either a short, or a missing ground, but in order fot a ground to cause that, you would have to have two problems at once. For that reason, I would look for wires that are either worn through the insulation, or melted together. It sounds like a pretty big short and to run the battery down quickly, it has to be large and also between the battery and any switch.
Hope this is of some help, or better yet that the problem is resolved and you are back on the road. If you read this, come back and let us know.