How do you balance writing new papers and revising drafts, especially when working on multiple projects with overlapping deadlines?

It is always the same! Take them by priority schedule and don't mixt it up.
Then things are going smoothly without problem. You can also delegate part of the work to your colleagues.

I maintain a prioritized task schedule that separates deep-focus blocks for writing new content from revision sessions, align each paper with its deadline and stage such as drafting, revising, submitting, and allocate dedicated weekly slots to revisit older drafts while keeping momentum on new writing—ensuring consistent progress across all projects without burnout.

When working on multiple projects and juggling time and efforts requierd to meet deadlines, I have always  set priorities  based on critical needs of the clients and delgated  certain tasks to my trustworthy staff who could  deliver promised work in a timely manner.

Revising drafts sometimes is more tedious that writing a new paper. I always delegate. My students write the drafts of new manuscripts and I revise the drafts.