We present a unified framework, called MorteX to treat embedded interface problems in the context of tying and contact. This framework is a combination of the X-FEM and mortar methods, whereby Lagrange multipliers are used to enforce interface constraints. As known, mixed formulations are prone to mesh locking which is characterized by the emergence of spurious oscillations in the vicinity of the tying interface. To overcome this inherent shortcoming, we suggest a new coarse-grained interpolation of Lagrange multipliers. This technique consists in selective assignment of Lagrange multipliers on nodes of the mortar side. The optimal choice of the coarse-graining spacing is guided solely by the mesh-density contrast between the mesh of the mortar side and the number of blending elements of the host mesh. The applicability of the proposed stabilization techniques is demonstrated for both the tying and contact problems solved within the MorteX framework.