Jettwhite,
I get the impression that the possible entrance/sinkhole is on your or your father's land. Is that correct?
I looked in my copy of "Caves of Tennesse" and did not find any "Dripping Cave," or anything even similar. I am sure there are some TCS members on this forum; they will have access to the data for TN and can help you figure out if it is or ever was on the listing. If the sink is filled in with rocks, it may be possible to open it back up, although it might require a backhoe. Good luck!
Trogman