Hi DigitalSplendid. Can you check your AWS's security group to make sure that SSH port 22 isn't limited to just your IP? You'll want to make sure it's either open (not safe) or include the IP address of your new server. Since you're going from one AWS instance to another, if they're in the same VPC and have a private IP address within the same subnet, you could save yourself the hassle with the security groups by using the private IP address instead for the connection.
Also, depending on your AWS instance, the username could be different so you'll want to check that out as well. Usually, by default, you will not be able to log into the root account itself via SSH (although using the console you can) which looks like is what you're doing). If I recall correctly, when I migrated out of AWS I used the normal ec2-user account just fine for the migration.
Just a few things to check.