If you select and start an overseer quest, and immediately zone afterwards, it doesn't seem to remember that you started a quest. It does seem to work if you wait awhile in the zone after starting an overseer quest though, but its probably several minutes of waiting you need to do. I also noted that finished quests takes quite awhile to allow you to claim the reward, and sometimes you have to zone before that "DBC-finish"-button is replaced with a reward button I play on AB, if that makes a difference(don't know if overseer is hosted locally on AB?)
Sometimes I log check Overseer and I have 4/5 Missions running when there were 5/5 when I went to bed. Add another mission and check back later after a rotation and claim the ones that are done. I'll claim one from 5/5 that were running, claim the reward from the Quest reward window and still have 5/5 missions running. They disappear with the rotation. It has nothing to do with changing to another character on the same account.