Collective adaptation, whether in innovation adoption, pro-environmental or organizational change, emerges from the interplay between individual decisions and social influence. Agent-based modeling provides a useful tool for studying such processes. Here, we introduce the biased-independence $q$-voter model, a generalization of the $q$-voter model with independence, one of the most popular agent-based models of opinion dynamics. In our model, individuals choose between two options, adopt or not adopt, under the competing influences of conformity and independent choice. Independent choice between two options is determined by an engagement parameter, inspired by earlier agent-based model of eco-innovation diffusion. When the engagement parameter equals $0.5$, the model reduces to the original $q$-voter model with independence; values different from $0.5$ break the symmetry between the two options. To place our study in a broader context, we briefly review asymmetric versions of the $q$-voter model proposed to date. The novelty of this work goes beyond introducing a generalized model: we develop the pair approximation (PA) for an asymmetric $q$-voter model and, for the first time, validate it on empirical organizational networks. Our results show that the interplay of social influence, independence, and option preference generates discontinuous phase transitions and irreversible hysteresis, reflecting path-dependent adoption dynamics. Surprisingly, the PA agrees well with Monte Carlo simulations on some empirical networks, even small ones, highlighting its potential as a computationally efficient bridge between individual decision-making and collective actions.