The reason the number grid takes 20 swaps is that it can be split into five cycles of 5 (for instance, 1,8,15,17,24 is a cycle). A cycle of n cells can be solved in n-1 swaps, so each of the cycles takes 4 swaps to solve.

In the letter grid, you want to find smaller cycles if possible. It turns out you can find four cycles of 2 letters each (where a single swap would solve both letters), three cycles of 3 letters, and two cycles of 4 letters, as coloured below. This will take 16 swaps in total.

To advance the solution, swaps should only be made within those minimum cycles (in the number grid, since 1 and 15 were in the same cycle, swapping them advanced the solution, without solving either letter in the swap, although it did change a 5-cycle into a 2-cycle and a 3-cycle).

In the letter grid, if any swaps were made within the 2-cycles or 3-cycles, letter would be solved, so we need to look only at the 4-cycles. Swapping one pair within each of the two 4-cycles as shown will turn each 4-cycle into two 2-cycles, advancing the solution without solving any letters, so the answer to the bonus question is 2.