Drop coalescence has been the focus of many studies because of its relevance to a broad range of engineering processes such as polymer blending where control of the microstructure is important. Nevertheless there remain many poorly understood aspects of this familiar problem. In general, drop coalescence involves forces that bring pairs of drops into close contact, as well as short-range molecular forces to rupture the thin liquid film that separates drop interfaces prior to confluence. Typically, the rate-limiting step for drop coalescence is associated with squeezing fluid out of the thin film, before molecular forces become significant. Drainage of the thin-film between drops with tangentially-immobile interfaces is a well-understood problem. The classical theory that describes this problem has been extended in an attempt to describe the behavior of drops with tangentially-mobile interfaces. However, there are important qualitative differences that have been overlooked in the current theories. Here, we present a study on the axisymmetric near-contact motion of drops with tangentially mobile interfaces. A long-time asymptotic analysis is presented for small-deformation conditions. Under these conditions the drops are nearly spherical, except in the near-contact region, where a flattened thin film forms. According to our analysis, a hydrostatic dome does not form in near-contact region at long times, in contrast to the assumption underlying all previous analyses of this problem. Instead, the shape of the film in the near-contact region results from the absence of tangential stresses acting on it. We show that the long-time behavior of the system is qualitatively different than previously predicted. According to our theory, the minimum film thickness (rim region) decays with time as h_m~ t^(-4/5) and the thickness at the center of the film decays as h_0~ t^(-3/5) which is a faster decay than predicted by prior analyses based on a hydrostatic dome.