A 45-year-old woman presents to an urgent care center reporting increasing pain and swelling in her right index finger over the past 48 hours. She recalls a minor puncture wound to the tip of the same finger while gardening three days prior. She also reports feeling feverish and generalized malaise. On examination, her temperature is 38.5 °C, heart rate 98 bpm, blood pressure 128/78 mmHg, and respiratory rate 16 breaths/min. Her right index finger is held in a position of slight flexion, with diffuse fusiform swelling extending along the entire digit. There is marked tenderness upon palpation along the volar aspect of the finger, and severe pain is elicited with passive extension of the digit. Erythema and warmth are noted extending proximally from the puncture site. What is the most likely diagnosis and the critical immediate management step for this patient?