You asked two questions. First it is possible to be honest and open with your employees without revealing information that is not appropriate to reveal. The earlier responses (above) address this issue quite well.
Once early in my management career when I felt senior management was not being honest and open, one of my early mentors pointed out: “The executive that is honest because honesty is the best policy is not honest. He is just very shrewd!” Honesty and openness (“transparency”) are prerequisites for trust.
Second, the distrust you observe is much more than “anxiety.” It will be devastating – and deeply embedded in the culture – if not effectively addressed, sooner rather than later. Distrust will not go away, It will move underground and contaminate all relationships between the employees and senior management.
Trust cannot be “earned.” It can only be “given.” It is a fragile gift. Trust is a precious, living gift to be nurtured, protected and honored. Creating the environmental ecosystem (working environment) where the employees again give senior management the gift of trust is an opportunity to create a more aligned, engaged and productive organization than existed before. “There are no mistakes, only lessons. No failures, only research.”