Background: Psychological distress is a common comorbidity in chronic obstructive pulmonary disease (COPD), yet its relationship with disease severity remains incompletely understood. This study aimed to assess depression, anxiety, and stress using the Depression Anxiety Stress Scales–21 (DASS-21) and to examine their distribution across COPD severity stages. Methods: This multicenter, cross-sectional observational study included 285 clinically stable COPD patients enrolled between 2023 and 2025. COPD severity was classified according to Global Initiative for Chronic Obstructive Lung Disease (GOLD) criteria. Multinomial and binary logistic regression models were constructed to identify independent predictors of COPD severity and clinically significant psychological distress, adjusting for demographic and clinical covariates. Bayesian independent sample analyses and ANOVA effect size estimates were additionally performed. Results: Smoking exposure was independently associated with advanced COPD stages (GOLD 4 vs. GOLD 1–3: aOR 1.05, p < 0.001), as was dyspnea severity (mMRC: aOR 14.66, p < 0.001). In multivariable models examining psychological outcomes, COPD severity was not independently associated with clinically significant depression (p = 0.899), anxiety (p = 0.460), or stress (p = 0.843). In contrast, symptom burden measured using the COPD Assessment Test (CAT) score was consistently associated with depression (aOR 1.133, p < 0.001), anxiety (aOR 1.179, p < 0.001), and stress (aOR 1.144, p < 0.001). ANOVA effect sizes across GOLD stages were small (η² ≤ 0.047), and Bayesian analyses provided moderate to strong evidence supporting minimal differences in DASS-21-R scores between severity groups. Conclusions: Psychological distress is prevalent across all COPD severity stages and is not independently determined by airflow limitation. Symptom burden rather than spirometric severity appears to be more closely associated with emotional outcomes.