Self-orthogonal codes hold significant importance in coding theory and cryptography, particularly due to their diverse applications in lattices and quantum error correction. Recently, researchers have constructed new families of self-orthogonal codes from specific classes of functions, largely driven by a novel self-orthogonality criterion in odd characteristic based on p-divisibility, introduced by Li and Heng in 2024 [18]. In this paper, following the latter approach, we explore linear codes derived from a quadratic function in the second generic construction framework. By explicitly evaluating the weight enumerators via exponential sums, we show that the resulting codes have four weights, are p-divisible, and, consequently, are self-orthogonal. Furthermore, we apply these results to construct classes of Linear Complementary Dual (LCD) codes, as well as linear codes with a fixed Hull dimension. We focus on low-dimensional Hull codes, as they play a pivotal role in determining the automorphism group of a linear code, efficiently checking permutational equivalence between codes, and constructing entanglement-assisted quantum error-correcting codes (EAQECCs).