I might suggest a pragmatic approach to developing your go to market plan. Have you already identified who your school district buyer and influencer personas are? Did some of these types participate in your beta program? Are their experiences approved for sharing via your marketing efforts (presuming they were positive and can suit your needs)?
Proof of performance, ideally with outcomes associated with critical district objectives (pain points) is critical for winning the attention and dialog you need with district/school leadership. K-12 leaders are typically risk-averse and often rely upon peer networks to validate potential decisions. You may be best served to launch locally, or in a defined region where you can build a reputation of success among that community and, then expand - leveraging state-level organizations related to your app/solution category to spread awareness.
If your app offers a 'freemium' or intro. level model and can benefit teachers/students, you may also want to consider employing a targeted 'bottom-up' - strategy to compliment your efforts at the district level? This approach has proven effective for several current K-12 category leaders particularly among academic-oriented apps that can benefit from rapid teacher adoption. Obviously, this approach depends significantly upon your business model and how you expect to monetize your customer relationships over time.