To address correlated source-load forecast errors and delayed reliability feedback for flexible load dispatch in high-penetration renewable distribution systems, a flexible load dispatch method embedding analytical reliability indices is proposed. Firstly, critical contingencies are screened using uniform design, and forecast-error correlations are represented through Cholesky decomposition. Analytical minimum-load-shedding functions are constructed by combining the modified stochastic response surface method with polynomial chaos expansion (mSRSM-PCE). Then, load shifting and shedding decisions are mapped to expected demand not supplied (EDNS), and a comprehensive model considering reliability risk, regulation cost, user comfort loss, and flexible-load capacity cost is formulated. Lastly, the method is validated using 50 source-load scenarios generated by a generative adversarial network. Compared with conventional PCE, mSRSM-PCE reduces the mean absolute error and root mean square error of EDNS by 30.38% and 11.82%, respectively, while limiting the absolute relative error of mean EDNS to 0.0329%. The recommended flexible-load capacity ratio is 8% under the baseline condition, reducing EDNS by 10.30%~80.52% across four typical scenarios. Sensitivity analyses confirm reliable performance under different objective weights and forecast-error correlations.