My read was that the man in the jail cell with the bloody mouth was Billy. The two women talking at the end of the episode mention that Billy ran into the back yard with a mouth that was dripping blood. And they sure did focus on his dripping mouth for a long time. Plus we know from Audrey scenes that Billy and Tina know each other and Tina is apparently the mom that was present for that scene with Billy
Furthermore, is that man (who is potentially Billy/in the cell) in the same state as Dougie? Maybe he’s not drunk. Maybe he was just repeating everything in the same way that Dougie repeats what he hears. That might explain why he went missing?