There's no oil in your exhaust system
There could be if the motor was burning oil really bad, but you'd be blowing blue smoke out the tail pipe - not the case here I assume

It could be a leak from somewhere else that drips onto the exhaust and runs down to the lowest point before dripping to the ground.

Black inside your tail pipe is normal, every car is like that.

I get drips from my tail pipe, enough to make me see where it was coming from. Turns out its the rear AC unit that drips water onto the pipe, then onto the ground. I noticed sometimes that these drips are 'dirty' like you mention. Its because carbon can dissolve in water, and the black around the edge of the tail pipe is carbon



So back to your issue, are the drips 'oily'? or just dirty water.