A low pressure re-absorber is integrated with a sulfur-rich solvent flash drum or a sulfur-rich solvent stripping column in a solvent acid gas removal process that provides for sufficient sulfur concentration for the downstream sulfur recovery unit. In another aspect of the invention, carbon dioxide containing or carbon dioxide rich gas streams that are at a lower temperature relative to a lean solvent stream are used to cool those lean solvent streams and then optionally to cool other process streams, to save energy consumption.