Old and Classical Japanese adjectives
Let's look just at the ones that end in ～い in modern Japanese, classed as 形容詞【けいようし】 in mainstream Japanese-language grammars. This includes the ～ましい adjectives you mention in your question, while excluding ～な adjectives (technically 形容【けいよう】動詞【どうし】) like 静【しず】か or 大胆【だいたん】.
The Japanese Wikipedia has a brief description of these in the 古典【こてん】日本語【にほんご】の形容詞【けいようし】の活用【かつよう】 section of the 形容詞【けいようし】 article. This explains that ～しい adjectives are called シク活用【かつよう】 (-shiku conjugation) adjectives, from the way these conjugate in the adverbial form, contrasting with ク活用【かつよう】 adjectives. Both of these had the 終止形【しゅうしけい】 or terminal (sentence-ending) form ending in ～し, making them hard to differentiate just based on the 終止形.
How the シク活用【かつよう】 adjectives and ク活用【かつよう】 adjectives differ
Apparently the シク adjectives often described emotional or subjective qualities, things like "fun" or "reliable", while the ク adjectives described more physical or objective qualities, things like "red" or "big".
This gets more to the meat of your question.
The Japanese Wikipedia article linked above doesn't really discuss derivation. Broadly speaking, after researching the etymologies of individual terms, it seems like many (most?) of the シク adjectives are derived from verbs, potentially from the causative form. Examples:
- 羨【うらや】む "to envy" → 羨【うらや】ます "to make someone envy" → 羨【うらや】ましい "having the quality of making someone envy": "enviable"
- 寄【よ】る "to draw close" → 寄【よ】らす "to make someone draw close" → 宜【よろ】しい "having the quality of making someone draw close" (the older sense): "desirable", then leading to modern "good"
- 疑【うたが】う "to suspect" → 疑【うたが】わす "to make someone suspect" → 疑【うたが】わしい "having the quality of making someone suspect": "suspicious"
I wonder if this might have developed as a straightforward shift in usage of the 連用形【れんようけい】 of the causative: //-asi// is the 連用形 ending for the causative, and it's also already the 終止形 ending for many of these シク adjectives.
Irregularities: vowel shifts
Some of these derivations involve as-yet-unexplained vowel shifts, where the //a// in the regular causative becomes //o// in the adjective. Yoru → yorasu → yoroshii is one such example. 頼【たの】む → 頼【たの】ます → 頼【たの】もしい is another.
Speculation: //a// for external qualities, //o// for internal qualities?
Poking around on my own, I've encountered various term clusters where there appears to be an ancient //a// ↔ //o// alternation. Broadly, the //a// variant seems like it describes some kind of outward or extrinsic quality, while the //o// variant describes an inward or intrinsic quality.
Examples seem to show up in both the first and second vowels of various word pairs:
- //has-//, the root underlying はさむ and possibly 端・橋・箸: describing a narrow distance between two things externally
- //hos-//, the root underlying ほそい: describing a narrow distance between the two edges of a thing internally
- たむ, earlier form of modern verbs たまる and ためる: something collects outwardly, like a puddle
- とむ, something collects inwardly, enriching a thing
- こまる, something gets stuck
- こもる, something goes into something else
- くら, "darkness", something that appears dark
- くろ, "black", something that is intrinsically dark
- つば, a swordguard, physically an outward protrusion
- つぼ, a basin, physically an inward protrusion
I wonder if the shift to //o// that can be seen in some シク adjectives might be part of this, suggesting a more intrinsic or internal quality.
Not all シク adjectives are traceable to verbs. Some シク adjectives come from reduplicated forms, like 長々【ながなが】しい or 馬鹿馬鹿【ばかばか】しい. Some come from mysterious roots: 楽【たの】しい is a good example. So far as I'm aware, there is no attestable verb たぬ.
That said, if a given シク adjective ends in -ashii or -oshii, and it's not obviously from a reduplicated root, chances are good that it might be from a verb. Look for a related verb form in the dictionary, based on the part before that -ashii or -oshii ending -- this could be a good opportunity to expand your vocabulary. :)