Is it permissible to continue in a forum where inappropriate signatures are prevalent, knowing that I am a general moderator and the administration fears a lack of member engagement if they are prohibited, and the forum owner technically assists me in supporting a forum under my responsibility?

If the signatures contain something forbidden by Sharia, such as images of 'awrah (private parts) or obscene phrases, then they are forbidden. The forum owner must prevent them. If he is unable to control the forum according to Sharia guidelines, he must close it. As for the questioner's work, if she has the authority to delete and prevent, she must do so. If she does not have this authority, she bears no sin for remaining, as long as she condemns the evil and does not assist in it.