Specification of monitored context properties and their influence on behavior of Web services and management activities is a prerequisite for contextsensitive operation, which is a characteristic management issue for mobile/embedded Web services. Our Web Service Offerings Language (WSOL) 1.1 provided formal specification of classes of service, different types of constraints, and management statements for Web services. In this paper, we present new WSOL version 1.4 that enables specification of monitored context properties, context monitoring schedules, context management parties, and context exchange mechanisms. It also enables descriptions of how context influences behavior of Web services, monitoring of Web services, and dynamic adaptation of Web service compositions. We verified feasibility of these improvements using syntax checks of the WSOL 1.4 grammar and modifications of prototype WSOL-related tools. We validated them on emulated case studies. An important novelty of this work is using context information in contract-based management of Web services.