A gas lift apparatus has a gas lift valve that disposes in a mandrel. A housing of the valve has a chamber, and a seat disposes between the inlet and outlet. A piston movably disposed in the housing has one end exposed to the chamber. A distal end can selectively seal with the seat to close the valve. A first edge-welded bellows disposed on the piston separates the inlet and chamber pressures and can fully compress to a stacked height when the distal end of the piston seals with the seat. A dynamic seal can be achieved at closing by using a captive sliding seal between the piston's distal end and the seat. A second edge-welded bellows can also be disposed on the piston, and the two bellows can operate in tandem. Oil filing the interiors and the passage can move from one bellows to the other to transfer the pressure differential between the inlet and the chamber pressures. The second bellows fully compresses to a stacked height and stops opening of the valve.