Free-running dual-comb systems based on a single laser cavity are an attractive next generation technology for a wide variety of applications. The high average power achievable by dual-comb thin-disk laser (TDL) oscillators make this technology especially attractive for spectroscopy and sensing applications in the molecular fingerprint region enabled by nonlinear frequency conversion. However, the high noise levels of TDL oscillators, e.g., induced by the turbulent water-cooling of the disk, are a severe challenge for spectroscopic applications. In this contribution, we confirm for the first time the suitability of dual-comb TDLs for high-resolution spectroscopy. Based on the novel concept of polarization splitting inside a TDL, our oscillator generates two asynchronous pulse trains of 240-fs pulse duration at 6-W and 8-W average power per pulse train and ∼97-MHz repetition rate at a central wavelength of 1030 nm. In the first detailed noise investigation of such a system, we identify the repetition frequency as the dominant noise term and show that ∼85% of the frequency noise of the comb lines of both pulse trains is correlated (integrated from 200 Hz to 20 kHz). We detect the absorption spectrum of acetylene in free-running operation within a measurement time of 1 millisecond. Being highly suitable for nonlinear frequency conversion, we believe the here presented result is an important step towards simple yet powerful mid-infrared dual-comb systems for high-resolution spectroscopy.