Integrated Security and User ID \ Password authentication are mutually exclusive. To connect to SQL Server as the user running the code, remove User ID and Password from your connection string:
$SqlConnection.ConnectionString = "Server = $SQLServer; Database = $SQLDBName; Integrated Security = True;"
To connect with specific credentials, remove Integrated Security:
$SqlConnection.ConnectionString = "Server = $SQLServer; Database = $SQLDBName; User ID = $uid; Password = $pwd;"