I appreciate the screenshots that you have added as that is very helpful! Could you let us know which theme you are using? As this context will help determine the next steps to take to have the footer links and logo reorganized.
If you are unsure about which theme is live on your store, you can navigate to the Theme Editor and select the three dots that are located towards the top left of the page. I took a screenshot from my test store to show you the section I am referring to:
Looking into the Dawn theme, we can move the blocks in the footer section by navigating to the Theme Editor. If you select on the footer section, you can mouse over each of the sections to see the option to drag and drop the sorting blocks. I took a screenshot from my test store to show you what I am referring to:
With that being said, if you notice that these changes are too limited for you - you can also make a change of coding to the theme to have the footer section display to your liking. Since the Dawn theme is designed and supported by Shopify, we can use free design time for Shopify themes to have our internal team make the changes on your behalf.
If you are interested in using the design time, you will need to login to your store using our Contact Us page. This will allow us to securely verify you as the store owner before speaking directly with a Support Advisor about the changes you would like to see.