Yes, you can use multiple load balancers for the same domain or different domains. For the AWS managed SSL certificate, you will need all domain names in the same certificate or you can use a different SSL certificate for each domain name. I would just create one certificate with "example.com" and "*.example.com".
Your issue is that you are using an SSL certificate for api.example.com that has the name example.com. The SSL certificate must include the name that matches the domain name.
If you do not have an SSL certificate installed on the EC2 instance, then setup SSL termination at the load balancer. This means creating a listener for port 443 that forwards to port 80.
To configure Route53 with an ELB in a different account:
For ELB load balancers, do one of the following:
- If you used the same account to create your Amazon Route 53 hosted zone and your load balancer – Choose Alias Target and choose a load
balancer from the list. If you have a lot of load balancers, you can
type the first few characters of the DNS name to filter the list.
- If you used different accounts to create your Amazon Route 53 hosted zone and your load balancer – Enter the value that you got
in the procedure Getting the DNS Name for an ELB Load Balancer.
- If you used one AWS account to create the current hosted zone and a different account to create a load balancer, the load balancer will
not appear in the Alias Targets list.
- If you used one account to create the current hosted zone and one or more different accounts to create all of your load balancers,
the Alias Targets list shows No Targets Available under Elastic
Load Balancers.
In either case, the console prepends dualstack. to the DNS name. When
a client, such as a web browser, requests the IP address for your
domain name (example.com) or subdomain name (www.example.com), the
client can request an IPv4 address (an A record), an IPv6 address (a
AAAA record), or both IPv4 and IPv6 addresses (in separate requests).
The dualstack. designation allows Amazon Route 53 to respond with the
appropriate IP address for your load balancer based on which IP
address format the client requested.
Values for Alias Resource Record Sets