The offshore-onshore regional integrated energy system (OORIES), which includes offshore gas production platforms and wind farms, along with the onshore gas-fired combined cooling, heating, and power supply campus, can effectively improve economic and environmental benefits by the complementary and coupling operation of multiple energies. In this paper, multi-energy coupling elements and multiple energy networks in the OORIES are modeled in detail. Considering the stochastic fluctuation of the offshore gas source and wind power output, a stochastic optimal dispatch (SOD) model of OORIES with multiple energy storages, including associated gas storage, hydrogen and heat storage tanks is proposed. Furthermore, a carbon trading mechanism is introduced to ensure the environmental benefits. An improved state-space approximate dynamic programming (SSADP) algorithm is proposed to solve the SOD model. By using the approximate value functions, the multi-period SOD model is decoupled into a series of single-period optimization models which are recursively solved period by period. The approximate analytical expression of the value function concerning states of each period is obtained based on the polynomial approximation, and combined with the transition probability matrices of stochastic quantities, an accurate and efficient solution for each single-period optimization model is obtained. A case study on an OORIES, and compared with the scenario-based algorithm, chance-constrained programming algorithm and traditional SSADP algorithm, demonstrates the effectiveness and high efficiency of the proposed method.