Change Primary Nav Search to 3rd Party Search

I'd like to use 'Advanced Woo Search' plugin to replace the default search and wondered what's the best way so the nav search icon uses this new search method?

I'm using the navigation as a header option with the search icon alongside primary nav. The search plugin can be inserted using <?php if ( function_exists( 'aws_get_search_form' ) ) { aws_get_search_form(); } ?> or via a widget.

Also, the search bar when opened isn't looking that great. I've applied a max-width to help but the close button doesn't sit within the search bar - it floats on top and covers the borders. On Mobile the search bar opens with the X almost in the middle as that's where the search icon is positioned in the nav bar.