How to improve on non-technical skills like writing Software design document?

First, I recommend these other career advice threads:

• "How can I write a great design doc?" - Here I go through the structure I use for my technical design doc.
• "How can I minimize thrash when I present technical designs for my projects?" - A common failure mode I've seen with mid-level engineers and proposing technical design in particular is that they're unable to efficiently lock in a decision. This Q&A goes through how to prevent that in-depth.

Here's some other thoughts from me:

• Look for examples - Copy-paste is one of the most powerful abilities in tech, and it applies here too. I assume there are senior/staff engineers on your team that write excellent design docs that you can easily understand. Use their docs as a template for yours. If you have questions (which you should), just ask them. If you're going to book an ad-hoc chat, you can say something like, "I'm trying to get better at writing design docs for my projects, and I really enjoy going through yours. Do you have 15-20 minutes to explain some of XYZ techniques you use - I would love to understand them and incorporate them into my doc for ABC project."
• Just do it a lot - The cool thing about design docs is you don't need some massive, high-stakes 6-month project to do it (for those, you want to be good at writing docs by then). Anything that takes >1 month can have a small design doc, and for smaller 1-3 week tasks, you can create a super "mini" design doc in the task ticket itself or in an RFC pull request.
• Read engineering blogs - Engineering blogs of top companies need to be very readable for maximum reach and brand impact, so they invest a lot of time making them concise and have incredible diagrams. Those are a great role model.

I also heavily recommend going through the System Design Series I made for a real-life example of what a thorough technical design doc looks like at a Big Tech company: System Design Masterclass: Taro Playlists