This paper proves several congruences for sums involving binomial coefficients modulo \(p^2\) for a prime \(p\). The main result is \[\sum_{n=0}^{p-1} \left(\sum_{k=0}^n \binom{n}{k} \frac{\binom{2k}{k}}{2^k} \right) \sum_{k=0}^n \binom{n}{k} \frac{\binom{2k}{k}}{(-6)^k}\equiv \left(\frac{3}{p}\right) \pmod{p^2},\] where \(\bigl(\frac{3}{p}\bigr)\) is the Legendre symbol; this generalizes a conjecture of Z.-W. Sun [Finite Fields Appl. 46, 179–216 (2017; Zbl 1406.11007)]. The proofs use hypergeometric function identities, the \(p\)-adic gamma function, and combinatorial identities proved by the Sigma software package.