With the mushrooming deployment of volatile renewable energy sources as well as the intrinsic uncertainty from the demand side, secure and economic energy dispatch has become increasingly challenging for energy systems, especially for the emerging and promising integrated electric-gas system. Aside from the aforementioned uncertainties, the dispatch of the integrated electric-gas system suffers from two inherent obstacles, which are model nonconvexities, originating from the Weymouth equations in the gas network, and the demand-side differentiated gas delivery priorities according to current industrial practice, respectively. To deal with the conundrum, an adjustable robust dispatch method is proposed for operating the integrated electric-gas system, where uncertain wind generation outputs and gas loads are described by intervals. In contrast to existing work employing pre-determined intervals, the admissible wind output intervals in this paper are optimized, reflecting the interdependencies between the regulation capabilities of gas-fired generation and the gas delivery adequacy. By this means, gas delivery priority is considered in comply with gas industrial practice, and it also provides a more flexible mechanism to maintain robustness of the dispatch strategy. Through analyzing the feasibility impact of uncertain variables, a deterministic robust counterpart is derived, in which uncertainties are eliminated based on affine generator dispatch and estimation of total line pack. Furthermore, nonconvex quadratic terms in the Weymouth equations are expressed as difference-of-convex functions. A sequential convex optimization procedure is developed, and a heuristic method is suggested to initiate the sequential algorithm. The proposed models and methods are tested on three test systems. Key impact factors on the dispatch strategy, such as gas prices, wind generation forecast accuracy and gas network initial operation conditions, are analyzed, and the computational benefits brought by convex programming are validated by scalability tests.