Edit : I had written this while wendy was posting... I think it’s better to first make a local install then once you have ported your old website to export your db, make a new install on your hosting service and then restore the db and local files.
Is your site powered by a content management system, or some kind of script ? Or do your pages contain php scripts ?
MODx can import static websites but it seems this is not your case, thus if you consider using MODx to run your website, you’ll have to import the current templates, stylesheets into MODx.
One way to do it smoothly is install MODx on a local server (using something like
XAAMP is the easiest way), then import the html structure of pages and turn them into templates, place css files in the assets/templates directory and start the adapting process.
If you run some kind of php script, you either will have to turn those into snippets, or check if MODx doesn’t have a snippet, plugin or module that could fit your needs.
Hope it helps