How do I cope as a manager caught between my team and leadership?

You stand in the middle, taking pressure from the chaos above and shielding your team below, and no one stands there for you. Everyone leans on you and you lean on no one. That is a lonely place, and the constant absorbing of everyone else's stress will hollow you out if you never set it down. Being the shield is part of the job, but being the shield with no relief is not sustainable, and it was never meant to be. You need somewhere to put the pressure too. Find a peer, another manager who lives the same squeeze, or a mentor, and be honest with them. Saying the hard part out loud to someone who understands takes real weight off your shoulders. You also do not have to absorb everything. Some of the pressure from above is not yours to carry alone, and passing along a clear, honest picture to your team is not failing to protect them. It is treating them like adults. Protect them from the chaos, yes, but do not protect them into thinking you are made of stone. Take care of the shield, or there will be nothing left to shield anyone with.