Oxy-fuel combustion was investigated in a model combustor that mimics the geometry of operational air-fuel gas-turbine combustors implementing the LPM DLE DACRS technology (Lean Premixed, Dry Low Emissions, Dual Annular Counter-Rotating Swirlers). Premixed swirl-stabilized CH4/O-2/CO2 flames were tested over ranges of equivalence ratio (phi = 0.2-1.0) and oxygen fraction (OF = 25-70%). The non-preheated flow rates of CH4, O-2, and CO2 were adjusted for each flame to maintain a common bulk velocity of 5.2 m/s at burner throat throughout the entire study. The combustor stability map was identified by quantifying the flashback and blowout limits within the phi-OF space. The adiabatic flame temperature (T-ad) was also mapped over the same ranges. Selected flames were imaged to analyze the effects of phi, OF, and T-ad on flame shape. Interestingly, at constant inlet bulk velocity, the flashback and blowout limits were found to follow contours of constant T-ad, which is a novel finding for oxy-methane flames. This indicated that T-ad is a more relevant parameter choice than phi or OF to quantify the combustor stability map. It was also found that a single correlation exists between flame speed and T-ad, with greater speeds at higher temperatures towards flashback, No similar correlation was found between flame speed and combustor power density, throat Reynolds number, or bulk mass flow rate. Flames of the same T-ad demonstrated almost identical shapes, which is another novel finding for oxy-methane flames. OF and phi showed very similar individual effects on flame shape, because both parameters can be represented by T-ad. Spanning the stable combustion zone from lower T-ad values to higher ones was observed to reduce flame size significantly.