If you or your co-worker is overworked, it is almost certain you will not have the energy or time to self-correct using feedback. Yes, there are exceptions, but people with the discipline to better themselves when under constant pressure are rare.
Taking on lots of work is a not a badge of honor. It is not a foolproof way to keep your job. While slackers are despised, those who are unreliable are even more despised.
Your boss and co-workers will have many demands on you, but that does not mean you need to agree and put pressure on yourself. Your task is to set the right expectations and manage them as you find the right sequence of projects to execute.
The first step is to provide transparency to what is going on in your world. Describe the problems you see, connect the dots and identify the dependencies between the problems and the teams that need to solve them. List the constraints and your suggestions for overcoming them. The outcome is to gain the participation and commitment of your co-workers to solve the problem.
The point being, you don’t have to be the hero, the caped crusader who saves the world. In fact, you can’t. But you can play a very key role in solving problems in your workplace, and hope someone notices. Sometimes the satisfaction of doing a job well is all you can hope for.