The low Earth orbit (LEO) satellite megaconstellation can provide ubiquitous coverage and high-performance connectivity, supporting multi-slice applications with various key performance indicator (KPI) requirements. However, due to the dynamic nature of LEO satellites, limited resources, and the diverse demands of different slices, managing user association (UA) and resource allocation becomes an increasingly challenging task in areas with overlapping satellite coverage. This paper proposes a joint optimization model for UA and resource allocation in multi-slice LEO satellite networks (SLSNs). Based on mixed-integer non-linear programming (MILP), our model minimizes the total propagation delay and optimizes the demand satisfaction ratio (DSR) using a Max-Min approach to ensure each slice meets its unique throughput requirements. In addition, a visibility-aware component is incorporated to prioritize longer satellite visibility, reduce handovers, and improve network stability. Due to the computational complexity of the MILP model, we propose a heuristic-based balanced association with delay-aware bandwidth distribution (B-DAD) approach. B-DAD operates in two phases: the initial UA phase selects satellites based on a combined metric of delay, load, and visibility duration, while the residual bandwidth distribution phase reallocates unused bandwidth among associated users proportionally. Extensive simulations demonstrate that our approaches significantly improve DSR, propagation delays, transmission delays, and network stability compared to the widely adopted benchmark maximum sum of data rate (Max-SR) method under varying elevation angles. Our findings highlight the effectiveness of the MILP model in achieving optimal solutions and the efficiency of B-DAD as a scalable alternative for large-scale scenarios.