The social icons in your primary menu are styled differently to the rest of the menu’s links, which is why I suggested the separate menus for your widget area.
As you’d like to include the social icons there, also, you could consider creating three menus:
1. Your primary menu that includes the main links as well as the social icons.
2. Your first widget menu that includes only the main links, not the social icons.
3. Your second widget menu that includes only the social icons, not the main links.
The last two menus could then be added to your footer via the custom menu widget and it would be easier to style them with CSS.