Background/Objectives: Autistic adults exhibit core and comorbid features that can have an impact on their daily functioning and lead to considerable psychological distress. Significant and consistent associations have been identified between autism characteristics—restricted repetitive behaviors and sensory features—and high levels of stress and anxiety. For a better understanding of the issue, it is necessary to consider the cognitive aspects that can help explain variations in stress and anxiety levels among adults with autism. We therefore aimed to model the contributions of worry and intolerance of uncertainty on the relationship between autism characteristics (sensory sensitivities and repetitive behaviors) and psychological distress (stress and anxiety). Methods: The sample comprised 144 autistic individuals with a mean age of 35.15 (SD = 11.44). They participated in the study by completing an online questionnaire to assess the study variables. Most of the participants reported being diagnosed with Asperger syndrome (63.6%) or Level 1 autism spectrum disorder (27.1%). Results: The model fit index values tested through path analysis indicated a good fit (χ2 = 5.65; p = 0.13 ns; CFI = 0.993; NFI = 0.985; RMSEA = 0.079; SRMR = 0.025) and identified worry and intolerance of uncertainty as significant mediating variables within a comprehensive explanatory model. Conclusions: These findings highlight the importance of worry and intolerance of uncertainty as specific targets in interventions aimed at improving stress and anxiety problems in autistic adults.