The global push for low-carbon electricity generation has made hydrogen-enriched natural gas an attractive near-term decarbonization option. This paper combines experimental and thermodynamic analyses of H₂–CH₄ combustion in gas turbine combustion chambers. Experiments were conducted on a patented two-stage swirl burner across 240 operating conditions. The effects of hydrogen fraction (γ = 0–40%), swirler vane angle (30°, 45°, 60°), equivalence ratio (φ = 0.17–1.00), and fuel injection strategy were measured against NOx and CO emissions and lean blowout stability. Each 10% increase in hydrogen content raised NOx by 23–24% via the Zel’dovich thermal mechanism, while CO fell by up to 28.5% at φ = 0.3 and 60° due to enhanced OH-radical activity. The minimum recorded NOx was 12.08 ppm (Type 2 injection, 30°, γ = 0%, φ = 0.3). Hydrogen addition improved lean blowout stability by 32–46% per 10% H₂. A parallel thermodynamic analysis showed that integrating an organic Rankine cycle (ORC) and supplementary H₂–CH₄ firing in the heat recovery steam generator cuts specific CO₂ emissions by 7.5–10% and raises net efficiency by 0.79–4.0 percentage points. Critical comparison with 28 published studies identified an optimal operating window: γ = 20–30%, φ = 0.5–0.7, 45° vane angle (SW = 0.8).