I started a new save (classic career) after the update and I am using the default skin.
I do not quite understand the last question. I can see the trn badge for players who will move in other clubs (or joining as free agent), but I cannot see the badge for those in my own team (when I checked the transfer page it was indicated that the player was moving though).
Normally I think I can get a message in the inbox if a player of my own team is approached by the other team (for end-of-contract transfer). If I remember correctly, the message should be something like "XYZ team has offered a contract to ABC".