The reason is that the "search space" used in each iteration is determined based on the number of eigenvalues requested. By increasing k, the size of the subspace is increased, which in this case means that all eigenvalues can be found much faster. There's no way of knowing the optimal subspace size to use, EIGS just uses max(2*k, 20) by default. So increasing opts.p will have the same effect as asking for more eigenvalues.
The reason EIGS is having problems is that the largest eigenvalues of matrix adj are very close together, and the algorithm used in EIGS depends on a gap between eigenvalues.
An alternative would be to just call eig(full(A)) and compute all eigenvalues and eigenvectors. For this specific matrix, it could be faster than EIGS, since the matrix is real symmetric and has a difficult eigenvalue distribution for EIGS. On my machine, this took 2.5 seconds, while most EIGS calls I tried took about 1.5 seconds.