Controlling the motion of non-Newtonian drops on surfaces is crucial for applications ranging from inkjet printing to biomedical devices and food processing. While the macroscopic behavior of viscoelastic drops sliding on tilted hydrophobic surfaces has been characterized—showing reduced velocities and elongation compared to Newtonian fluids—the underlying microscopic mechanisms remain poorly understood. To address this gap, we developed a high-speed, high-resolution reflection microscope that enables direct visualization of the contact line of sliding drops. We used water-soluble polyelectrolyte solutions based on polyacrylamide and allowed drops to slide on hydrophobic substrates composed of Teflon amorphous fluoropolymer and polydimethylsiloxane-coated glass slides. The substrate tilting angle was varied between 20° and 45°. We reveal how viscoelasticity influences the dynamics of the receding contact line and drop motion. Our experiments demonstrate that viscoelasticity can destabilize the receding contact line, triggering filament formation. This instability previously observed in the coating of thin viscoelastic films is reported here for sliding drops, for the first time on smooth surfaces. We further highlight the critical role of polymer charge in this process: while cationic and nonionic polymers promote filament formation, anionic polymers do not, a difference we attribute to the distinct wetting properties of the solutions. In conclusion, we clarify the interplay between rheology, surface interactions, and drop dynamics.