Forward queries to public DNS servers if Private Hosted Zone doesn't have specific record

You cannot do this. If the zone apex is the same for private and public zones (domain.com and domain.com), if the record is not in the internal zone, it will return NXDOMAIN. From the documentation.

If there's a private hosted zone name that matches the domain name in the request, the hosted zone is searched for a record that matches the domain name and DNS type in the request, such as an A record for accounting.example.com.

Note:

If there's a matching private hosted zone but there's no record that matches the domain name and type in the request, Resolver doesn't forward the request to a public DNS resolver. Instead, it returns NXDOMAIN (non-existent domain) to the client.

It's look like route table configuration issue. Create 2 subnets public and private than in route table edit Subnet associations. like public route table has public IPs and private has private IPs and give internet gateway to public route table. It will resolve this issue.