The various pud- words from Latin pudīcitia will likely be recognizable only to English readers who are also speakers of Modern Romance or scholars of Ancient Latin, or both.
(Well, perhaps except for pudenda, but I can’t guarantee that folks will know why it means what it means; folk etymologies abound.)
For alternatives, “a sense of modesty, shyness, bashfulness, reticence, decency, chastity, shame, shamefulness” are all in common use and some may well apply, depending on your intent.
From the OED:
Frequency (in current use): Band 2 contains words which occur fewer than 0.01 times per million words in typical modern English usage. These are almost exclusively terms which are not part of normal discourse and would be unknown to most people. Many are technical terms from specialized discourses. Examples taken from the most frequently attested part of the band include decanate, ennead, and scintillometer (nouns), geogenic, abactinal (adjectives), absterge and satinize (verbs). In the lower frequencies of the band, words are uniformly strange or exotic, e.g. smother-kiln, haver-cake, and sprunt (nouns), hidlings, unwhigged, supersubtilized, and gummose (adjectives), pantle, cloit, and stoothe (verbs), lawnly, acoast, and acicularly (adverbs), whethersoever (conjunction).
About 45% of all non-obsolete OED entries are in Band 2.
Origin: A borrowing from Latin. Etymon: Latin pudentia.
Etymology: < post-classical Latin pudentia modesty (5th cent. in Augustine) < classical Latin pudent- , pudēns pudent adj. + -ia -y suffix3; compare -ency suffix. Compare earlier pudicity n.
Modesty, bashfulness, or reticence; embarrassment; an instance or expression of this.
For pudicity meaning shamefulness (so the opposite of impudicity meaning shamelessness), the OED gives:
Etymology: Partly < Middle French pudicité modesty, chastity, behaviour which shows a sense of shame (1417; French pudicité ; < pudique pudic adj. + -ité -ity suffix, after classical Latin pudīcitia), and partly < classical Latin pudīcitia sexual purity, chastity < pudīcus pudic adj. + -itia -ity suffix. Compare Italian pudicizia (a1292).
Modesty, decency; chastity.
And also places pudicity in the same Frequency Band Two as it places pudency, a band for fairly rare words in English that most people will not know. So most people would understand neither of those two lovely Latin loanwords lost to a time when any educated English speaker was also literate in at least Latin if not also something like French or Italian. This is no longer true.
That means it won’t do you any good to tease out some nuance of difference between them, or with any of the several other terms ultimately deriving from the same Latin etymon, like pudor, pudic, pudify, pudibundery, pudendous, pudor, pudent, pudeur. Those are all much too “fancy” for anything except the most technical of texts, preferably one that gets to define its own terminology.