Davey, there’s definitely something quirky going on, though I can’t tell immediately what it is. Off hand it looks like there’s an HTML tag that may not be closed somewhere. The plain vanilla version on my server isn’t having some of the same issues I’m seeing on yours, most of which appear to stem from the formatting of the left hand sidebar.
Are you running a child theme? I know you’ve made some customizations in at least a few places. Perhaps try starting from scratch and then re-adding your modifications one at a time to see which may be causing the conflict?
Whatever is wrong is not affecting your archive pages (eg: https://daveymoloney.com/2020/02/12/ or https://daveymoloney.com/kind/article/) which appear as expected. This means it’s unlikely an issues with changes you made to your header, footer, sidebar, or archive files. Did you make any direct changes to content, single, or index.php files?
You might also try removing some of your widgets temporarily to see if there’s a conflict with one of those, though I’d guess that’s less likely unless you have one or more that appear on some pages and not others.
Whatever is causing the issue should fix almost all of the wonky display issues you’re seeing to give you the “sticky note” effect you’re looking for. If you want to make other simple changes, you can browse through what’s available in the Customizer, some of which is described here: https://wordpress.org/support/article/twenty-fifteen/.