Repulsive and attractive hydrodynamic interactions are measured between an oil droplet (n-hexadecane) and various glass microspheres in aqueous environments with an atomic force microscope (AFM). The magnitude and form of the hydrodynamics in a spherically wrapping thin film are investigated with a parametric study on external approach velocity and probe radius. The actual sphere-drop separation at closest approach can be experimentally deconvoluted in some situations from parametric data without ambiguity. Theoretical force profiles are calculated from the augmented Young-Laplace equation modified to allow for two distinct hydrodynamic drainage regimes: Reynolds lubrication for thick films with slightly deformed drops and a wrapping film condition for the indented interface. Also, an increase in film stability is directly observed as a function of velocity between a hydrophobized glass sphere and oil in pure water and in the presence of sodium dodecyl sulfate (SDS) below the critical micelle concentration (CMC).