If you see your rankings dropping, chances are you have some backlinks working against you. I wouldn't go hog wild and disavow everything, as the link profile could have some spots that help you. I can't be specific without looking at it.
The best thing you can do, in my opinion, is spread the good news about the new site and redesign, and not "build" links from it, but "earn" links. Share your awesome new content and get people to like it so much that you earn some new traffic and links. You are on the right track by learning how to do it right (versus the old, shoddy SEO company that didn't do much good), but now you have to put the work in and do it right.
There are plenty of good methods that work for getting you some good, organic, quality traffic that will help your SERPs. If you feel you are over your head, there are plenty of people on here that would be more than willing to help you out.