This problem is much easier to solve. Usually, unhappiness due to overwork has a root in problems that you can address. For example, if overwork is due to (in)stability of the production systems, it’s your job as the manager to slow down the product roadmap in order to focus on stability for a while. Make clear measures of alerts, downtime, and incidents, and strive to reduce them. My advice is to dedicate 20% of your time in every planning session to system sustainability work (“sustainability” instead of the more common “technical debt”).
In a case where overwork is due to a pressing, time-critical release, remember two things. First, you should be playing cheerleader. Support the team however they need supporting, especially by helping out with the work yourself. Order dinner. Tell them you appreciate the hard work. Make it clear that they’ll have explicit break time after the push. Make it as fun as you can in the moment. Sometimes a crunch period can serve as a bonding experience for a team. But they’ll remember whether their manager was with them during the stressful period, or off somewhere else, doing her own thing.
Second, do everything you can to learn from this crunch period and avoid it the next time. Cut features if you can. Push back on the date if it’s truly unrealistic. Crunch periods will happen, but there is no reason they should happen frequently.