Learn English – How to better express the idea of “written communication”

Relevance is the more common form, according to grammarist.com:

Relevance vs. relevancy:

• There is no difference between relevance and relevancy. Though the latter is the older form, relevance is now preferred in all varieties of English. In this century, relevance is about ten times as common as relevancy in U.S. popular usage, and the gap is even wider in British, Australian, and Canadian sources.

Ngram relevance vs relevancy

Etymology:

• relevance (n.) 1733; see relevant + -ance. Related: Relevancy (1560s).

-ancy (suffix):

• word-forming element denoting quality or state, from Latin -antia, forming abstract nouns on past participle adjectives in -antem, appearing in English mostly in words borrowed directly from Latin (those passing through French usually have -ance or -ence; see -ance).

• Not all adjectives have a natural adverbial use. I'd think you'd be hard pressed to use beautiful where beautifully is called for.
• In "She is small," small is an adjective serving as a nominative complement. It's not an adverb.
• It might be best to consider participles as verb forms that have uses as modifiers. When they modify nouns, that's an adjectival use, but it really doesn't make them adjectives. An adjective (say, red) may be compared (to get redder). That won't work for cooked. You'll have to say "more cooked."
• I don't have time to search for a past participle that can't have an adjectival use. Most can.
• In "the book was written", written is part of the past passive; it's not a modifier. You can tell because you can sensibly append a prepositional phrase with by to give who wrote the book. Similarly, "to be cooked" might be passive infinitive if you're concerned about who did the cooking. Same test. If you're concerned with the internal temperature, then it would be a participial modifier.
• Past participles have adverbial (or at least semi-adverbial) uses: "He lived cursed by fate." The participle would say (at least in part) how he lived.