Contradicting my previous comment, I now think that both Playfair codewords are solvable without a Playfair cracker, though the longer one may be easier. In the case of the latter it can be established from three of the letter pairs that EBD must appear in a line or a column, as must IDM. QE in row 1 is also helpful. After some experimenting I had the skeleton of a grid with four letters of the codeword in place. I then entered the pattern into Chambers. Ignoring words with repeated letters there were only two options, differing by one letter.
The shorter codeword is harder because the encoding of DI doesn't result in either of those letters so they are clearly not in the same row or column. It's still doable.
Admittedly I already knew the codewords, but I set that aside and simply experimented with moving known letter arrangements around.
I rather regret running out of patience and resorting to the online aid. I'm pretty certain that the editors would not have allowed the puzzle had an online code cracker been essential.