Contemplated configurations and methods include a solvent regenerator (58) that has an upper (93) and a lower stripping section (94). Cooled rich solve is used as reflux while heated rich solvent (11) is used as a source of stripping agent in the upper section (91). A reboiler (62) in the lower section provides further stripping agent, hi especially preferred configurations, a portion of lean solved from the regenerator (58) is further stripped in a separate or integrated regenerator (62) to form an ultra-lean solvent. Both lean and ultra-lean solvents are preferably used in a two-stage absorber (52) to thereby from the rich solvent and an offgas that is very low in acid gas.