Dr. Lee,
The search visibility dropping is likely due to a higher-traffic keyword phrase ranking or two dropping your website off of page 1 of Google. I am not sure what keyword phrases' rankings you are tracking with Moz Pro, but I am assuming it would be directly due to that. There is talk of a Google algorithm update going on as I write this in May 2017, but there are not many details on it yet (and Google Fred may have had an effect as well depending on when you first noticed the rankings drop). I would make sure your website content is up-to-date, and written to be helpful to your potential patients with each page having unique, non-spammy ads or duplicate content on it, and then setup Google Search Console and Google Analytics on your website, if you have not already, as this will have more specific details for you to analyze over time.
As for as the "Links" aspect of SEO, this can be a controversial topic, but generally any website wanting you to pay for a backlink, is probably not worth having a link to your website on (generally speaking). There are a lot of great directories that will help establish your new office more and aid in your NAP (Name, Address, Phone Number) Listing consistency, which can help your Google Maps rankings. Think BBB, Yelp, Healthgrades, Angieslist, etc. (Moz Local is a great tool for checking this). Having a monthly blog post on your website with relevant topics and helpful content is also a great way to increase your Search Visibility. Here is also a great guide by Brian Dean on Link Building in 2017- http://backlinko.com/link-building. Hope this helps and best of success!