I've encountered that when I was implementing a project for the company. I was the only point of contact and a lot of things were dependent on me being able to make the product work. I didn't like it, but I just had to put more work in. A lot of overtime work, studying the documentation, talking to different teams. It was a matter of making the most of one's time. I can't be waiting on a person's availability, I need to be doing something while they are not available. It was really tough work. But in the end it was successful, and it was the reason I was promoted.
There's many practical ways one can find efficiency, productivity and reduce stress, and there's many emotional ways to deal with the stress...the trick is knowing, and using, them at the right time.
Well done on your promotion...gained through hard work, consistency, effort and other such things no doubt. Earned, not given.
Couldn't have said that better. Thanks.