Hi claybrain! I think you might be interested in Gloom, a card based game that has a few expansions that can allow you to have 6 players. The goal is to give your family a terrible life through event cards, which give you a negative score, and happy events to your opponents family members, giving them positive points. After one family is completely killed off, whoever has the most miserable dead family members is the winner. The game has interesting artwork and encourages story telling for how each event came to happen to the family members.
The Resistance is always really fun too. If you’ve ever played Werewolf, it’s kind of similar. Your group is the resistance but there will be some among you that are assigned to be spies. You’ll need to try to figure out who the spies are before you fail too many missions and the spies win.
For a more complex cooperative game, I suggest Space Cadets. Each player will be assigned the duties of a specific officer on board your ship. Through teamwork you’ll work towards the goal which changes and can be to gather space crystals or combating alien ships.
Hope this helps!