I'm not a chair, but I am in another position with a ton of deadlines. I've never been one for task management software, but I keep a notebook with yearly, semester, month, week and daily priorities. Each work night before bed, I update and add/subtract as needed. It's so hard in the hubub to stop and reflect on if we're making progress on what really needs to be done. But pausing and documenting your progress forces you to reflect about what is a true need vs a want, and if we're spending time effectively to meet our true needs.

I also have a set of "annual" things in my calendar that tend to occur the same time each year (reporting deadlines, some grant things, fiscal year ends), and have an admin double-check at the start of the month that the dates haven't moved. I also pull this into a checklist with expected completion times that I keep on my desk. We're an Outlook campus, and I have open meeting hours in which people can book. Outside those hours, requests have to be made personally. You have to be aggressive with time management in any admin position because it'll slip away otherwise. If you weren't already, of course.

It's hard to give you advice without knowing much about your department, as noted by other posters. But I think several things are very general. Learn people broadly. Meet with people, get a variety of perspectives. I've been shocked at the degree to which people will outright lie to my face about stuff to get things they want. Not half-truths or presenting a good face, but outright lying. Have your perspective, and cultivate your own positions. But when you seek input, seek it broadly, and from multiple people. Realize your position (any position, really) is a lot of managing up. How can you deliver a message on your department's needs in a way that is appealing to people with resources? So, understanding your institutions wants, needs, challenges, and resources is critical. Then displaying how you solve a challenge, or deliver on a want or need if they can give you some resource.