The rapid uptake of natural gas-fired units in energy systems poses significant challenges in coordinating the electricity and gas systems. Besides, the uncertainty caused by integrated renewable energy such as wind power raises more requirements on the robustness of the operation for integrated electricity and natural gas system (IEGS). To address these challenges, this paper investigates the distributed adjustable robust optimal power and gas flow (OPGF) model for IEGS. Using linear decision rules (LDRs), we first propose an improved adjustable robust model combining with the automatic generation control systems to fully exploit its potential in dealing with renewable energy uncertainty while utilizing the controllable polyhedral uncertainty set to reduce solution conservatism. This improved LDRs based adjustable robust approach can reduce the computational burden caused by the existing decomposition based robust approach when applied to distributed optimization. Then, to preserve the information privacy and decision-making independence of subsystems, two tailored alternating direction method of multipliers (ADMM) based distributed optimization frameworks for IEGS with and without a central coordinator are presented, respectively. Effectiveness is illustrated through benchmark case studies.