We study the necking and pinch-off dynamics of liquid droplets that contain a semidilute polymer solution of polyacrylamide close to overlap concentration by combining microfluidics and single DNA observation. Polymeric droplets are stretched by passing them through the stagnation point of a T-shaped microfluidic junction. In contrast with the sudden breakup of Newtonian droplets, a stable neck is formed between the separating ends of the droplet which delays the breakup process. Initially, polymeric filaments experience exponential thinning by forming a stable neck with extensional flow within the filament. Later, thin polymeric filaments develop a structure resembling a series of beads-on-a-string along their length and finally rupture during the final stages of the thinning process. To unravel the molecular picture behind these phenomena, we integrate a T-shaped microfluidic device with advanced fluorescence microscopy to visualize stained DNA molecules at the stagnation point within the necking region. We find that the individual polymer molecules suddenly stretch from their coiled conformation at the onset of necking. The extensional flow inside the neck is strong enough to deform and stretch polymer chains; however, the distribution of polymer conformations is broad, and it remains stationary in time during necking. Furthermore, we study the dynamics of single molecules during formation of beads-on-a-string structure. We observe that polymer chains gradually recoil inside beads while polymer chains between beads remain stretched to keep the connection between beads. The present work effectively extends single molecule experiments to free surface flows, which provides a unique opportunity for molecular-scale observation within the polymeric filament during necking and rupture.