The adverse road surface condition has been identified as an important factor resulting in serious casualties and property losses in traffic accidents, and there is a tremendous need to uncover the interaction mechanism between deteriorating road surfaces and vehicle impact locations on the driver injury severity at a disaggregate level. In this paper, three groups of random parameters logit models with heterogeneity in means (and variances) are developed to investigate the unobserved heterogeneity and temporal stability of the determinants affecting driver injury severity outcomes across different damage locations among single-vehicle crashes that occurred under adverse weather conditions. A three-year crash dataset gathered from January 1, 2015, to December 31, 2017, in Ohio is utilized. Three crash injury severity categories including no injury, minor injury, and severe injury are identified as outcome variables, while crash characteristics, driver characteristics, temporal characteristics, vehicle characteristics, roadway characteristics, and environment characteristics are regarded as potential predictors influencing driver injury severities. Additionally, likelihood ratio tests and marginal effects are used to assess the temporal instability and impact location non-transferability of the explanatory variables. The results indicate an overall temporal and locational instability of model estimates while several determinants are identified to have consistent effects on injury severity outcomes such as animal-involved collisions, old drivers, safety restraint usage, female drivers, physically impaired drivers, and vehicles with insurance. This study also quantifies and characterizes the net effect of year-to-year and location-to-location shifts through probability differences between out-of-sample predictions and within-sample observations. Varying magnitudes and inconsistent directions of distribution characteristics (mean, skewness, kurtosis, and prediction accuracy) in the probability differences across different impact locations over time are captured. Moreover, this study indicates that the non-transferability of collision locations has a greater impact on the prediction accuracy than the temporal instability. The findings could potentially serve as a reference for transportation administrators to formulate effective safety strategies to better protect drivers from adverse-road-related crashes.