Aerial physical interaction represents a promising direction for next-generation unmanned aerial vehicles (UAVs), but it requires an aerial platform that can exert contact forces while maintaining stable flight. For close-proximity tasks, this translates into three coupled design requirements: multidimensional wrench generation for stable contact, compactness for maneuverability and safety in confined spaces, and reduced lateral airflow toward the target when generating horizontal force. This article presents FLOAT Drone, a fully actuated coaxial UAV with servo-driven control surfaces for close-proximity physical interaction. The coaxial dual-rotor layout provides a compact propulsion layout, while the control surfaces, immersed in the rotor downwash, generate lateral forces and moments for 6-DoF wrench generation. A force-matched computational fluid dynamics (CFD) comparison with a tilted-rotor alternative quantifies the reduction in target-facing lateral airflow. To account for nonlinear rotor--control-surface coupling in the rotor wake, a high-fidelity polynomial aerodynamic wrench model is identified from precision force measurements and embedded in a constrained nonlinear allocator for real-time wrench tracking. Comparative flight and interaction experiments show that the proposed framework improves control accuracy over linear allocation baselines, rejects ground-effect and payload disturbances, and enables close-proximity drawer push--pull manipulation through a $2~\mathrm{cm}$ handle clearance.