If you have a TA quasigroup you can simply rearrange the columns in such a way that the 0 are on the main diagonal. Then the quasigroup is (in general) a WTA quasigroup and can be used for the Damm algorithm. I've done this for order 32, see result below, and this is possible for every order except 2 and 6.
I think the quasigroup of order 10, which can be found in Wikipedia, is constructed by Lemma 5.2 of Damm's theses. This is because it should still detect the phonetic errors after rearranging the columns, so the elements need to be renamed and the rows need to be rearranged accordingly.
Finally, here is the WTA quasigroup of order 32 for the Damm algorithm:
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 03 01 07 05 11 09 15 13 19 17 23 21 27 25 31 29 02 00 06 04 10 08 14 12 18 16 22 20 26 24 30 28 01 03 05 07 09 11 13 15 17 19 21 23 25 27 29 31 04 06 00 02 12 14 08 10 20 22 16 18 28 30 24 26 07 05 03 01 15 13 11 09 23 21 19 17 31 29 27 25 06 04 02 00 14 12 10 08 22 20 18 16 30 28 26 24 05 07 01 03 13 15 09 11 21 23 17 19 29 31 25 27 08 10 12 14 00 02 04 06 24 26 28 30 16 18 20 22 11 09 15 13 03 01 07 05 27 25 31 29 19 17 23 21 10 08 14 12 02 00 06 04 26 24 30 28 18 16 22 20 09 11 13 15 01 03 05 07 25 27 29 31 17 19 21 23 12 14 08 10 04 06 00 02 28 30 24 26 20 22 16 18 15 13 11 09 07 05 03 01 31 29 27 25 23 21 19 17 14 12 10 08 06 04 02 00 30 28 26 24 22 20 18 16 13 15 09 11 05 07 01 03 29 31 25 27 21 23 17 19 16 18 20 22 24 26 28 30 00 02 04 06 08 10 12 14 19 17 23 21 27 25 31 29 03 01 07 05 11 09 15 13 18 16 22 20 26 24 30 28 02 00 06 04 10 08 14 12 17 19 21 23 25 27 29 31 01 03 05 07 09 11 13 15 20 22 16 18 28 30 24 26 04 06 00 02 12 14 08 10 23 21 19 17 31 29 27 25 07 05 03 01 15 13 11 09 22 20 18 16 30 28 26 24 06 04 02 00 14 12 10 08 21 23 17 19 29 31 25 27 05 07 01 03 13 15 09 11 24 26 28 30 16 18 20 22 08 10 12 14 00 02 04 06 27 25 31 29 19 17 23 21 11 09 15 13 03 01 07 05 26 24 30 28 18 16 22 20 10 08 14 12 02 00 06 04 25 27 29 31 17 19 21 23 09 11 13 15 01 03 05 07 28 30 24 26 20 22 16 18 12 14 08 10 04 06 00 02 31 29 27 25 23 21 19 17 15 13 11 09 07 05 03 01 30 28 26 24 22 20 18 16 14 12 10 08 06 04 02 00 29 31 25 27 21 23 17 19 13 15 09 11 05 07 01 03 03 01 07 05 11 09 15 13 19 17 23 21 27 25 31 29 00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 01 03 05 07 09 11 13 15 17 19 21 23 25 27 29 31 02 00 06 04 10 08 14 12 18 16 22 20 26 24 30 28 07 05 03 01 15 13 11 09 23 21 19 17 31 29 27 25 04 06 00 02 12 14 08 10 20 22 16 18 28 30 24 26 05 07 01 03 13 15 09 11 21 23 17 19 29 31 25 27 06 04 02 00 14 12 10 08 22 20 18 16 30 28 26 24 11 09 15 13 03 01 07 05 27 25 31 29 19 17 23 21 08 10 12 14 00 02 04 06 24 26 28 30 16 18 20 22 09 11 13 15 01 03 05 07 25 27 29 31 17 19 21 23 10 08 14 12 02 00 06 04 26 24 30 28 18 16 22 20 15 13 11 09 07 05 03 01 31 29 27 25 23 21 19 17 12 14 08 10 04 06 00 02 28 30 24 26 20 22 16 18 13 15 09 11 05 07 01 03 29 31 25 27 21 23 17 19 14 12 10 08 06 04 02 00 30 28 26 24 22 20 18 16 19 17 23 21 27 25 31 29 03 01 07 05 11 09 15 13 16 18 20 22 24 26 28 30 00 02 04 06 08 10 12 14 17 19 21 23 25 27 29 31 01 03 05 07 09 11 13 15 18 16 22 20 26 24 30 28 02 00 06 04 10 08 14 12 23 21 19 17 31 29 27 25 07 05 03 01 15 13 11 09 20 22 16 18 28 30 24 26 04 06 00 02 12 14 08 10 21 23 17 19 29 31 25 27 05 07 01 03 13 15 09 11 22 20 18 16 30 28 26 24 06 04 02 00 14 12 10 08 27 25 31 29 19 17 23 21 11 09 15 13 03 01 07 05 24 26 28 30 16 18 20 22 08 10 12 14 00 02 04 06 25 27 29 31 17 19 21 23 09 11 13 15 01 03 05 07 26 24 30 28 18 16 22 20 10 08 14 12 02 00 06 04 31 29 27 25 23 21 19 17 15 13 11 09 07 05 03 01 28 30 24 26 20 22 16 18 12 14 08 10 04 06 00 02 29 31 25 27 21 23 17 19 13 15 09 11 05 07 01 03 30 28 26 24 22 20 18 16 14 12 10 08 06 04 02 00 