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Single-cavity dual-comb mode-locked lasers, an emerging dual-comb technology, can generate two sets of pulse sequences with excellent mutual coherence, thus eliminating the need for complex active noise reduction systems. We report the generation of dual-wavelength dispersion-managed solitons in a near-zero anomalous-dispersion Er-doped mode-locked fiber laser. Transmissive dual channels are provided by birefringence filtering effect. The dual-wavelength mode-locking centers at 1539.1 and 1578.8 nm with the repetition-rate difference of 420 Hz. Furthermore, 2-hour spectral stability is measured in an open environment, and we analyze the intra-cavity dynamics that influences the long-term stability of the dual-wavelength mode-locked laser operating in the free-running state.