A construction of a topologically transitive mixing subshift with a fully supported invariant measure, but no fully supported ergodic measure, is given by Benjamin Weiss in the article "Topological transitivity and ergodic measures", Mathematical Systems Theory 1971. It gives a direct combinatorial construction, with $n=2$.
It is easy to give an example of a topologically transitive subshift on two symbols with no fully supported measure: just take the orbit closure under the shift of the sequence with all negative entries equal to one, and all non-negative entries equal to two.
