Integrated terrestrial and non-terrestrial networks (ITNTNs) are envisioned as a key paradigm for sixth-generation (6G) wireless systems, enabling seamless global connectivity. In this paper, we investigate a dual-aerial active reconfigurable intelligent surface (ARIS)-assisted non-orthogonal multiple access (NOMA)-based ITNTN, where a terrestrial base station (TBS) and a satellite (SAT) simultaneously serve terrestrial and satellite users with the aid of a UAV-mounted ARIS and a HAP-mounted ARIS. Users are multiplexed via power-domain NOMA with a predefined SIC decoding order. We formulate an average sum-rate maximization problem by jointly optimizing transmit beamforming, ARIS coefficients, and the 3D trajectories of the UAV and HAP, subject to power, unit-modulus, ARIS power, and mobility constraints. The problem is highly non-convex due to coupled variables, nonlinear SINR expressions, ARIS amplification, and trajectory-dependent channels. To address this, a block coordinate descent (BCD)-based framework is proposed. Specifically, beamforming is optimized via WMMSE, ARIS phase shifts via a manifold-based RCG method, amplification factors via SCA, and trajectories via first-order approximations. The proposed algorithm is guaranteed to converge to a stationary point. Simulation results demonstrate that the proposed design achieves significant performance gains over benchmark schemes. In particular, it provides an average sum-rate improvement of approximately $8.44\%$ over passive RIS under given power constraints, highlighting the benefits of dual-aerial ARIS and joint communication-mobility optimization.