Network Load Balancers, Elastic IPs, and Fargate Services + Tasks?

Hi, I'm new to AWS and trying to get some of our on-prem VMs migrated over to Fargate. From what I've read, I need to use a Network Load Balancer since I want to setup an Elastic IP on the NLB which can be delegated via an A record from our on-prem DNS (Route 53 isn't an option right now).

In ECS, I have a Task Definition for my container, which runs Nginx, listening on TCP 80 and 443. Also in ECS, I've setup a Cluster and two Services; One Service for TCP 80, and another Service for TCP 443. Each of these services have at least one Task defined which spins up the Nginx container.

When I define the NLB, I create two Listeners/Target Groups. One for TCP 80 and the other for TCP 443. When defining the Listeners, I assign the EIP to each listener. The NLB has an A record associated with it named something like my-net-lb-xxxxxxx.elb.us-east-1.amazonaws.com. There are also two public IP addresses associated with the A record but here's where it gets weird.

If I do an nslookup on NLB A record, I only see one of the EIPs attached to it. Likewise, I can point a browser at that EIP and I can reach the Fargate Task. The second IP seems to be orphaned somewhere. It's setup as a listener, but doesn't show up anywhere except under the other AZ for the NLB. Pointing a browser at that orphaned IP goes nowhere.

I was expecting a Round Robin DNS setup on the NLB A record so if one of the EIPs is unroutable because a Task is down, the client has a chance to refresh and roll-over to the other IP address.

I must be missing something. I've tried finding specific documentation on this setup but everything I've found so far is either incomplete, or so old it's irrelevant.

Thanks for any insight, and apologies this is so long