The Learn-As-you-Go (LAGO) design is an adaptive clinical trial design that allows modifications to multicomponent intervention packages across stages. Centers participate in more than one stage, as is common in large-scale implementation trials. In LAGO trials, center characteristics may act as confounders, predicting both the intervention package and the outcomes. We extend the LAGO theory by introducing fixed center effects to control for confounding by indication through measured and unmeasured center characteristics. Conditioning on center characteristics by including fixed center effects ensures asymptotic results hold without requiring explicit characterization of unmeasured confounders. Our methods apply even with small numbers of centers. LAGO theory is established for continuous outcomes following a generalized linear model and binary outcomes following a logistic regression model, unifying theory across outcome types. Point- and interval estimators are derived, and consistency and asymptotic normality are established. Valid hypothesis tests for the overall intervention effect are provided, and the optimal intervention package minimizing cost subject to a target outcome mean is obtained via constrained optimization.