Make a small kerf with a knife and a chisel, saw in that.

Look up Paul Sellers on YouTube, he has great hand tools videos.

If you're doing this with hand tools don't underestimate the importance of sharp chisels. From your description I don't think they're nearly sharp enough.

https://taytools.com/collections/drill-press-sanding/products/3m-775l-cubitron-ii-stickit-5-psa-sanding-discs

For anyone who struggles with sharpening a carpenters pencil, this should help a lot!

Card scrapers are incredibly versatile, making them an essential part of my surfacing kit for any project. Learn how to sharpen card scrapers, here.

By Asa Christiana #226–May/June 2012 Issue

• ceramic alumina wheels
• diamond dresser