The answer has to be, there is none that are still running (with the name ending in the letter e.
You might like to have a look at this link about off-Broadway theatres.
Below is a link for the longest running Broadway Shows:
As Wikipedia states; if the show is a hit, it often gets moved to a Broadway theatre.
If you scroll down this link you will find long running "Off Broadway" shows but it doesn't say whether they are still running or not.
It just has the number of performances.