Probabilistic linguistic preference relation (PLPR) provides an effective and flexible tool with which preference degrees of decision-makers can be captured when they vacillatingly express linguistic preference values among several linguistic terms. Individual consistency and group consensus are two important research topics of PLPRs in group decision making (GDM). Considering the problems associated with these two topics, this study proposes a novel GDM framework with consistency-driven and consensus-driven optimization models based on a personalized normalization method for managing complete and incomplete PLPRs. First, existing limitations of the traditional normalization method for probabilistic linguistic term sets (PLTSs) managing ignorance information are specifically discussed. Given the potential valuable information hidden in PLTSs, a personalized normalization method is newly proposed through a two-stage decision-making process with a comprehensive fusion mechanism. Then, based on the proposed normalization method for PLTSs, consistency-driven optimization models that aim to minimize the overall adjustment amount of a PLPR are constructed to improve consistency. Moreover, the developed models are extended to improve consistency and estimate the missing values of an incomplete PLPR. Subsequently, a consensus-driven optimization model that aims to maximize group consensus by adjusting experts’ weights is constructed to support the consensus-reaching process. Finally, an illustrative example, followed by some comparative analyses is presented to demonstrate the application and advantages of the proposed approach.