For taking a snapshot of an AWS EC2 instance and resetting the password, please take a look at How to take snapshot / image of AWS EC2 instance for creating new servers
After creating EC2 instance from AMI image for SQL server, you should change the Registered Servers for the SQL Server instance.
If not, the new EC2 instance will still have a registered server from the AMI image (from your old server).
Open SQL Server Management studio and under views, go to registered servers. (see below screenshot)
Under Local Server Group, expand it and you will find the old server name.
Right click on the old server and click properties.
In the server name dropdown, select your new server name and modify the name of the Registered server name. (see below screenshot).
Click Test and Save the settings.
You should be now able to connect to the SQL Server database instance, make sure all your sql server (non-windows) users are working as expected.