Faced a similar issue, the Service was unable to find the ".SETTINGS" file, this file usually exists in :
C:\Packages\Plugins\Microsoft.AzureData.WindowsAgent.SqlServer\VERSION\RuntimeSettings\1.Settings
In my case, it was in:
C:\Packages\Plugins\Microsoft.AzureData.WindowsAgent.SqlServer\1.1.2769.203\RuntimeSettings\1.Settings
And for some reason, the SqlServerExtension.Service.exe Service was trying to load an older version 1.1.2735.199.
You can identify which version is causing the error by stopping the service "Microsoft Sql Server Extension Service" , then going to "C:\Program Files\Sql Server Extension" using CMD and running SqlServerExtension.Service.exe., when doing so, it should point out which version it is trying to load, see this screenshot:
As you can see, it is trying to load from Agent Version ending with .199 while the latest is 203, I copied the files from 203 to 199 folder and it worked!