I know one says an act of desperation, but I've heard desperacy much more than I've ever heard desperation, it's like I've almost never heard desperation.
Why exactly was desperation preferred over desperacy and is there a pattern to noun adjectives ending with -ate?
I wonder where the OP has heard desperacy much more than desperation.
The OED says:
desperacy, n. Obs.
Etymology: < desperate adj., n., and adv.: see ‑acy suffix.
Desperateness, desperation.
- 1629 J. Gaule Practique Theories Christs Predict. 11
Downe to the nethermost depth beyond recouerie: Let vs there take our portion of desperacie.
which suggest that it was a used word, about two centuries ago. Ngram can find no example, nor can Google Books, nor can COCA.
It is not that that desperacy is not an English word. It is not a current English word.
Desperacy is “not a word”1 because just plain despair alone plenty suffices as a noun, without any redundantly tedious nominalizationalizing suffixes and postfixes and add-on-the-enders.
The take-away lesson here is that sometimes, perhaps even often, it is far better to trim things off instead of tack them on.
That is, not a word attested2 by the Oxford English Dictionary.
That is, not a word attested as being current in contemporary English.
However, the OED does attest an obsolete word †ˈdesperacy,
and for which it provides no citation later than from the 18th century.
As for the second part of your question, here are some attested triples that may interest you:
