The participation of wind farm-energy storage systems (WF-ESS) in electricity markets calls for an integrated view of day-ahead offering strategies and real-time operation policies. Such an integrated strategy is proposed here by co-optimizing offering at the day-ahead stage and operation policy to be used at the balancing stage. Linear decision rules are seen as a natural approach to model and optimize the real-time operation policy. These allow enhancing profits from balancing markets based on updated information on prices and wind power generation. Our integrated strategies for WF-ESS in electricity markets are optimized under uncertainty in both wind power and price predictions. The resulting stochastic optimization problem readily yields optimal offers and linear decision rules. By adding a risk-aversion term in form of conditional value at risk into the objective function, the optimization model additionally provides flexibility in finding a trade-off between profit maximization and risk management. Uncertainty in wind power generation, as well as day-ahead and balancing prices, takes the form of scenario sets, permitting to reformulate the optimization problem as a linear program. Case studies validate the effectiveness of the strategy proposed by highlighting and quantifying benefits w.r.t. other existing strategies.