"We are experiencing elevated API error rates and network connectivity errors in a single Availability Zone."
Key fact "A SINGLE AZ". Availability Zone's are isolated from each other with redundant power supplies and internet connectivity and most often physically different datacenter locations. Well architected applications are designed to allow for a single AZ to become unavailable. This is precisely why the cloud is useful: you can bring up new capacity in the other Availability Zones behind the same load balancer with zero effort - the autoscaling does that for you automatically.