June here from Shopify.
Thanks so much for including those screenshots! In a lot of themes, we can expect the main menu to re-format as a sidebar menu when it reaches a certain capacity. I can see you've already made great use of the space allowed in your menu, with nice short titles.
Beyond this, if you are hoping to keep your menu formatted horizontally across the top of your homepage (I can definitely see the appeal!), you might consider using drop-down menus to achieve this. Are there any other opportunities to group any of your current menu items into broader categories? This could definitely help you earn back some space. Although it looks like you already have a great handle on how these work, you can refer to our help document for any extra guidance or please let me know how I can help!
May I ask what theme you are currently using? If it is one of our free Shopify themes, our theme support team may actually be able to assist you with you the changes you're describing, as part of our complimentary design time. I'm also happy to explore any tutorials that are available for your theme so that you're aware of all of your options - Shopify theme or not.