Random generation with cycle type restrictions

We study random generation in the symmetric group when cycle type restrictions are imposed. Given $\pi ,{\pi }^{\mathrm{\prime }}\in S_n$, we prove that $\pi$ and a random conjugate of ${\pi }^{\mathrm{\prime }}$ are likely to generate at least $A_n$ provided only that $\pi$ and ${\pi }^{\mathrm{\prime }}$ have not too many fixed points and not too many $2$-cycles. As an application, we investigate the following question: For which positive integers $m$ should we expect two random elements of order $m$ to generate $A_n$? Among other things, we give a positive answer for any $m$ having any divisor $d$ in the range $3\le d\le o(n^{1\mathrm{/}2})$.