Learn French: L'hôtel était complet, alors nous avons dû partir.

In the Passé Composé, devoir shows that an obligation was actually fulfilled: we had to leave, and we did. If we used the Imparfait (nous devions partir), it would translate closer to 'we were supposed to leave', which leaves it ambiguous whether we actually did. Here, the completed action of leaving was a direct result of the hotel being full.

Why is the first verb était in the Imparfait when the rest of the sentence is in the Passé Composé?

This sentence perfectly shows how the two past tenses work together. The Imparfait (était) sets the background scene or state (the hotel being full). The Passé Composé (avons dû) is then used for the specific, completed action that resulted from that state (we had to leave).

Why does the past participle dû have a circumflex accent?

The accent on dû is only there to distinguish it from the mandatory contraction du (meaning 'some' or 'of the'). Because this accent is just a visual differentiator, it disappears when you add agreement endings, such as the feminine due or plural dus, where no confusion is possible.

Why is complet used here instead of plein for 'full'?

While plein means 'full' in a physical sense (like a glass full of water or a box full of items), complet is the standard French term used for hotels, theaters, trains, and restaurants to mean 'fully booked' or 'no vacancy'.