Ask for a custom website design
Open Menu, then WEBSITE, then My website, and find Custom design.
Fill in "Describe what you want" with the look and feel you are after. You can add "Inspiration links (optional)", which takes links to sites or photos you like, and you can also attach inspiration images.
Tap Submit request. The panel shows what a build starts at, so read the figure on screen rather than one you were told earlier.
There is no published turnaround for a quote. If you need it by a date, say so in the request rather than assuming one.
This is the studio doing the work, not a template switch. It is worth writing a few honest sentences about what you do not like about your current site, not just what you want.