One technique that has gotten traction in my organization is the notion of "email bankruptcy" and "slack bankruptcy". I.e. if you find your email or slack debt getting too big to handle, you simply declare bankruptcy, archive everything (or everything older than a day or two), and start fresh. If you are very far behind, the downsides of doing this seem to be no worse than the alternatives.
When I shifted from IC engineering work to management I found that I gradually became overwhelmed with email and other comms from a lot of people who now either wanted my opinion or felt I needed to be informed about a lot of things I previously wouldn't have been involved in. Oddly, but maybe not surprisingly, I found that there was positive correlation between the size of my email backlog and how crowded my calendar was, and that the causality seemed to operate in a feedback loop. If I had too many meetings it got harder to keep up with email, and if I didn't answer emails, I got invited to more meetings. Practicing near-zero inbox actually gave me more time, instead of less.