"Transversal" is a good old geometry word, a noun, as you say. It goes way back to long before anybody was thinking of transversality in the modern sense.
It grates on me to hear it used as an adjective, and this owes something to the fact that in my impressionable youth I saw one of the chapter-heading quotations in Hirsch's graduate text on differential topology: From Whitehead, "'Transversal' is a noun. The adjective is 'transverse'". No doubt this also had an impact on others who (like me) tend to be fussy about language.
On the other hand, language does drift along, and there's no stopping it, and generally no harm is done. By the time you perceive a serious need to tell the world that some usage is wrong, a case can always be made that it is no longer wrong.
In the case at hand it's understandable that "transversal" has come to be used an adjective; after all, "-al" looks like an adjective ending. (But there are words in English where people have been fooled by that, changing the language. "Bridal" is an example.)
By the way, if we were going to be sticklers on this point, mightn't we want to go back and change "transversality" to "transversity"?