I'm trying to figure out creative time management strategies concerning the "week" system. If I think my heros might get knocked out in a side quest could I abandon it to prevent morale loss? Is it even possible to "run away"?

Yep, just hit the surrender button. You will gain some gold and fame based on the run.

You can surrender and you will gain less gold and less fame than if you won but keep in mind that all the search tokens you flipped are computed independently of the quest awards.

Surrendering does not affect your morale.

You still lose the 1 week by traveling to this quest in the first place. So you can only save some morale by surrendering before everyone gets knocked out

-zxql

It actually worked out well. We had a week before we had to do the main quest so we wanted to maximize our gold and fame but head into the main quest with our morale topped off. We abandoned the side quest (Hook Line & Tentacle) before we got knocked out and got 30+ gold and some fame. Funny enough I think we only got 50 gold from Burning Wrath the main quest which we won.