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If you are concerned about "link juice" following to the new pages, you will need to set up 301 redirects.  I  don't know much about web.com but some of these services don't have a way to 301. It looks like some of your pages have different urls now but Google still has the index from the old site. The pages that have exactly the same names will be OK. If there is no "link juice" to worry about, you may be all right just waiting for Google to figure it out. They will eventually re-index the new site. You might also consider a different solution for your site using a CMS like wordpress and self hosting. That may give you more flexibility in the long run and a blog on your site would be good as well. Do a little reading here as there is a great deal of good advice. Congratulations on finding MOZ!

If you're using Wix.com, and they use AJAX... here's what you can do to verify google is scraping your site properly. go to google.com and simply type "site:yourURL.com"... no spaces between the colon and your site name.  This will return Google's index of your site, and then you can look at the cached pages to ensure they're seeing what they should. Wix posted about this, and they provide both the needed AJAX (end user) option and the HTML (google crawler option) properly..  here's their blog from 2015 addressing this (see #4) https://www.wix.com/blog/2015/04/myth-busting-10-interesting-facts-about-wix-seo/ if you're using Wix.com... or other website builder tools... make sure to submit your sitemap to the Google Search Console, this should also help.