Latin Squares whose transversals share many entries

We prove that, for all even $n\ge 10$, there exists a latin square of order $n$ with at least one transversal, yet all transversals coincide on $\lfloor n\mathrm{/}6\rfloor$ entries. These latin squares have at least $19n^2\mathrm{/}36+O(n)$ transversal-free entries. We also prove that for all odd $m\ge 3$, there exists a latin square of order $n=3m$ divided into nine $m\times m$ subsquares, where every transversal hits each of these subsquares at least once.