Hi Wonderpants,
In the beginning two characters were bugged but this was fixed in the first patch, character should not have an effect on any player now as it is purely cosmetic. I think a lot of issues reported mention the same character as they are some of the more common choices. That being said the team are told about characters when bugs are reported just in case. I have not heard of this issue being related to character choice from the team. Every time you start a game, the game asks you to pick a skin, this skin is then applied to all saves, you can change this back any time from options in the main menu.
I am sorry this continues to happen for you. Unfortunately before the original fix was issued the team released a comment saying that if it happens once in a save it is likely to happen again. It appears to occur when the save info for a client is not correctly called from a host. A common factor others have mentioned is a gap between playing, the longer the gap, the more likely it'll occur. Have you tried swapping around hosts to a host with a stronger upload speed to see if this helps? while not ideal it may be something that can help while the team works to resolve this issue.
Clare