Hey,
Using a second domain and starting fresh would be the best idea. However, don’t just get any old domain, get a second, good domain you can use for the business.
This way when it ‘goes live’ you won’t need go through wordpress admin, and the database changing all instances of the new domain, which you would have used throughout the site for images, links etc. Then simply point the original domain to the new server.
And, as you are completely changing the site, it would be ideal to add 301 redirects from the old page urls to the new equivalent pages on the new site. So you don’t loose any existing SEO benefits.
Regarding new hosting, if you haven’t got any in mind, I can recommend United Hosting, they have great support too!
Hope this helps.