The theme is in charge of the font and its weight. Trying to change that per page could make a mess, especially if you change fonts or change themes.
In the editor, the sidebar will show the Advanced section when you have a block selected. This is for a class, which is a name, not CSS. Pick a name like semi-bold and type that in the Advanced field. Then define that class in the Customizer > Additional CSS as.semi-bold { font-weight: 600; }
Don’t overlook the leading dot, which is needed in CSS for class names.
You should see your change in the preview. If you don’t, some other CSS could be more specific and overriding your rule. Once you get it working, you can put that same class name in the Advanced field, on all the blocks you want to affect.
@joyously that worked perfectly! Thanks so much for clearing that up for me.
