The designer doesn’t provide ready access to the Current link state, but it does exist. Here’s how I get around it:
Head to your home page in the designer.
Add a new link and link it to your home page. (We’ll delete this later, don’t worry.)
Check the style panel and see that this link now has a Current class selector applied.
Style however you’d like this Current state to look, and then delete the link.
As far as the menu appearance not being correct inside the designer, would be happy to look into this as well, but I’d need a read-only link to your Webflow project to take a look. Instructions are here. Thanks!