There is no risk here. You don't need to take the doxy.
To get an STI, you need to have oral, vaginal or anal sex. With oral sex, it's not the saliva that's a concern, but the exposure to the throat. If your penis goes into a throat, it is exposed to whatever germs reside there. STIs reside in the throat.
Oral sex on a vagina is a lot lower risk than oral sex on a penis because nothing goes into the throat.
Even if something has a risk - this did not - a few seconds or a brief touch isn't going to be enough time to transmit anything.
Also, the term is transgender. not ladyboy.
The doxy you took, though - it's called DoxyPEP, post exposure prophylaxis. It's a new thing, and can work to prevent syphilis, gonorrhea and chlamydia in people who were assigned male at birth. (It does not work in people who were assigned female at birth.) It needs to be taken within 3 days of the exposure.
Think of it as the morning after pill, but for certain STIs.
You should only take it after an actual risk, because resistance can develop, and you should still do your regular testing.
That's all just for information - you had no risk here. :)