Can I see the full due date on a closed card?

Yes!
By default, cards display the number of days left until their due date. But to see the exact date, just hover over the calendar icon on the card.

Want the full due date to be visible at a glance on all cards with the due date set? Enable the Task Summary power-up from the board settings, and set it up to show due dates:

If you don’t see a due date field on your cards, head to Settings → Card Template and turn it on. Keep in mind that board settings are only accessible to project managers or board owners.