Unless I hear from the tenant as to when the rent is coming, we'll serve a pay or quit notice as soon as the grace period has expired.

In court cases I've attended (listening to others evict for nonpayment those seem to be the most common reasons folks are in court) the local judges tend to focus on whether a landlord provided a 3 day Pay or Quit notice to a tenant late on rent (although ironically Georgia landlord/tenant laws only state that a landlord can begin the process immediately after demanding rent from a tenant who then does not pay). 

My partner and I noticed there are a lot of properties in Greenville, NC that are vacant and owned by the city.

It also states that there is a $500 "re-rent fee" charged to all tenants who have not completed their full term, or who do not give proper 30 days notice.

I’ve also noticed a very high proportion of travel nurses live with a dog or two, I know it’s a risk for you as the landlord, but being pet friendly will definitely appeal to travel nurses.

With not having confirmation on certified mail being delivered, how would we move forward with pay or quit notices or eviction notices?

Point them in that direction and meanwhile tell them you understand if they need to find another place to live and you will let them out of the lease with 30 days' notice (ie you can rent it for Dec 1).

I know I've noticed several on Craigslist here in Texas.

@Dominic Kosteris currently having difficulty with my inbox I didn't notice till now

Foremost is our Go-To carrier also, but there are some others out there BUT the replacement cost limits will be noticeably lower.