Dallas here from the Shopify Social Care Team and I are happy to see what I can do to help you with this customization.
By the looks of what you're wanting I think you might want to checkout the Minimal theme. That one has the ability to have your announcement, checkout and search bar all in-line at the very top of the page. With the Supply theme, it will be broken up as it is showing in the image that you shared.
However, your announcement banner has a bit of a white gap at the top of the page. I went to check my test theme and that gap didn't show up for me so if you decide to stick with the layout in the Supply theme I would love to take a closer at your store to see what I see on your store from my side. Are you able to share your store url?