(1) Background: Consumer switching behavior was one of the expected outcomes of the ongoing competitive reform in the European electricity industry. The aim of this study is to analyze the factors that affect the intention of consumers to switch from their incumbent supplier, who has a strong market position. This article follows the trend of describing and analyzing the situation of incumbent energy suppliers after energy market liberalization. (2) Methods: The research goal was to verify seven research hypotheses regarding switching intentions on the Polish energy market. For this purpose, structural equation modeling analysis using AMOS 29 was conducted. The survey was carried out in 2020, before the SARS-CoV-2 pandemic in Poland, and the research sample consisted of 1216 adults (clients of the incumbent supplier). (3) Results: Special attention was given to customers’ resistance to change, as it was found to be a factor that can impact customers’ decisions to switch, both directly and indirectly. Resistance to change can affect the trust that customers have in their current supplier’s brand, their perception of the costs involved in switching, and their intention to switch. The research also highlighted the importance of brand trust and perceived ease of change in the customers’ decision-making process. (4) Conclusions: An important factor that reduces switching behavior is the customer’s resistance to change; brand trust is a valuable asset that also reduces switching intention.