Proper etiquette after applying for a job - email HR or not?

Hi all, since I’m now past the halfway point for my student teaching semester and my college is having their career fair in a little over a week, I decided to hop on the local job board for all of the school districts in my area and see what was already posted. We’re still in early March so it was, understandably, not much, but one district near me that’s towards the top of my list of places I’d want to teach and is opening 2 new schools for the new school year had a ton of open positions listed. I applied for one of the high school social studies positions listed.

An hour or so later I went on their website (I think I was going to look at their pay scale once bonuses and coaching is factored in) and ran across their HR page, and found the publicly available emails for their Director of Human Resources for secondary schools and the Chief Human Resources Officer. I went ahead and fired off a quick professional email to each of them as a follow up, introduced myself, background, qualifications, all that and attached my resume and cover letter.

After I sent the emails I thought whether or not that’s the right move? Any advice on whether or not that’s the thing to do?