Continuous casting (CC) is one of the most important processes of steel production; it features a high production rate and close to the net shape. The quality improvement of final CC products is an important goal of scientific research. One of the defining issues of this goal is the stability of the casting process. The clogging of submerged entry nozzles (SENs) typically results in asymmetric mold flow, uneven solidification, meniscus fluctuations, and possible slag entrapment. Analyses of retained SENs have evidenced the solidification of entrapped melt inside clog material. The experimental study of these phenomena has significant difficulties that make numerical simulation a perfect investigation tool. In the present study, verified 2D simulations were performed with an advanced multi-material model based on a newly presented single mesh approach for the liquid and solid regions. Implemented as an in-house code using the OpenFOAM finite volume method libraries, it aggregated the liquid melt flow, solidification of the steel, and heat transfer through the refractory SENs, copper mold plates, and the slag layer, including its convection. The introduced novel technique dynamically couples the momentum at the steel/slag interface without complex multi-phase interface tracking. The following scenarios were studied: (i) SEN with proper fiber insulation, (ii) partial damage of SEN insulation, and (iii) complete damage of SEN insulation. A uniform 12 mm clog layer with 45% entrapped liquid steel was additionally considered. The simulations showed that parasitic solidification occurred inside an SEN bore with partially or completely absent insulation. SEN clogging was found to promote the solidification of the entrapped melt; without SEN insulation, it could overgrow the clogged region. The jet flow was shown to be accelerated due to the combined effect of the clogging and parasitic solidification; simultaneously, the superheat transport was impaired inside the mold cavity.