This paper offers a thorough analysis of the coverage performance of Low Earth Orbit (LEO) satellite networks using a strongest satellite association approach, with a particular emphasis on shadowing effects modeled through a Poisson point process (PPP)-based network framework. We derive an analytical expression for the coverage probability, which incorporates key system parameters and a distance-dependent shadowing probability function, explicitly accounting for both line-of-sight and non-line-of-sight propagation channels. To enhance the practical relevance of our findings, we provide both lower and upper bounds for the coverage probability and introduce a closed-form solution based on a simplified shadowing model. Our analysis reveals several important network design insights, including the enhancement of coverage probability by distance-dependent shadowing effects and the identification of an optimal satellite altitude that balances beam gain benefits with interference drawbacks. Notably, our PPP-based network model shows strong alignment with other established models, confirming its accuracy and applicability across a variety of satellite network configurations. The insights gained from our analysis are valuable for optimizing LEO satellite deployment strategies and improving network performance in diverse scenarios.