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UPDATE TableName
SET DBTextField = REPLACE(CAST(DBTextField AS varchar(MAX)), 'SearchText', 'ReplaceText')
FROM TableName
WHERE CHARINDEX('SearchText', CAST(DBTextField as varchar(MAX))) > 0


Example:
UPDATE [database].[dbo].[table]
SET [column_name] = REPLACE(CAST([column_name] AS varchar(MAX)), 'OLD_WORD', 'NEW_WORD')
FROM [database].[dbo].[table]
WHERE CHARINDEX('OLD_WORD', CAST([column_name] as varchar(MAX))) > 0