I would not delete your old Google Search Console account. You can keep it around for troubleshooting purposes. First of all, it houses historical data. Second, if you see activity on it when there shouldn't be any, as unlikely as that is, you know there's an issue somewhere.
There's no special reset you need to perform with GSC. Just get your site set up in GSC the way you normally would: verify your ownership, set up your www and non-www versions and choose your preferred one, submit your sitemap. (I also advise doing the same in Bing Webmaster tools, so you can rank better there too, and submit your sitemaps and monitor for issues.) The one extra thing you may want to do is use the change of address tool in GSC, to signal to Google that your site has moved.
Outside of any webmaster tools application, you still need to make sure that your old website is properly redirecting to the new website: that you have a solid 301 redirect scheme in place, that your old pages are not appearing in SERPs (and that they're not resolving - they have to redirect, or go to 404 in some cases), that you're not seeing any serious traffic to your old site in Google Analytics (use a new tag for your new site), etc. Basically, work to get your old site out of the index, and for your new site to benefit from the old one's authority and link profile.