A 38-year-old woman presents to an urgent care clinic with a 2-day history of escalating fever, profound sore throat, and increasing difficulty and pain with swallowing solids and liquids. She reports her voice has become muffled, and she feels generally unwell. Three days prior, she was evaluated in an outpatient setting for similar but milder symptoms, where rapid strep and viral pharyngitis tests were negative, and she was advised symptomatic management. Her vital signs are: temperature 38.8 C, blood pressure 122/80 mm Hg, pulse 95/min, and respirations 18/min. She exhibits a tripod position, appears anxious, and occasionally clears her throat. Her oropharyngeal examination reveals only mild erythema with no exudates, swelling, or obvious tonsillar enlargement. Given these findings, what is the most appropriate immediate diagnostic step and the potential serious complication if definitive management is delayed?