Latin Squares whose transversals intersect in unusual ways

A latin square of order $n$ is an $n\times n$ array in which each of $n$ symbols occurs exactly once in each row and column. A transversal in such a square is a selection of $n$ entries that includes one representative of each row and column, and one of each symbol. For all even orders $n\ge 28$ except $n=30$, we construct a latin square of order $n$ in which every pair of transversals share at least one entry. We conjecture that in our squares there is no single entry that is common to all transversals. We prove this conjecture for $n\le10\,000$ by finding transversals using an algorithm that is likely to be of independent interest.
We say that a transversal is dominant if it intersects every other transversal of the same latin square. We show that there exist latin squares of order $n$ that have a dominant transversal for $n\in\{5,7\}$ and also for all $n\ge8$ such that $n\not\equiv3\bmod4$.