Modern social networks rely on recommender systems that inadvertently amplify misinformation by prioritizing engagement over content veracity. We present a control framework that mitigates misinformation spread while maintaining user engagement by penalizing content characteristics commonly exploited by false information, specifically, extreme negative sentiment and novelty. We extend the closed-loop Friedkin-Johnsen model to incorporate the mitigation of misinformation together with the maximization of user engagement. Both model-free and model-based control strategies demonstrate up to 76% reduction in misinformation propagation across diverse network configurations, validated through simulations using the LIAR2 dataset with sentiment features extracted via large language models. Analysis of engagement-misinformation trade-offs reveals that in networks with radical users, median engagement improves even as misinformation decreases, suggesting content moderation enhances discourse quality for non-extremist users. The framework provides practical guidance for platform operators in balancing misinformation suppression with engagement objectives.