ea Rare, for /ɛː/ (see ee).
ee /eː/, becoming [iː] by about 1500; or /ɛː/, becoming [eː] by about 1500. In Early Modern English the latter vowel came to be commonly written ⟨ea⟩. The two vowels later merged.
oa Rare, for /ɔː/ (became commonly used in Early Modern English).
The surmise by @sumelic is what I've always assumed:
the spelling arose simply because /ɛː/ is halfway between /a/ and /e/ in vowel quality. It seems parallel to <oo> and <oa>
The fact that what was an occasional Middle English spelling variant became prevalent in Early Modern English (just before it merged out of phonetic reality) suggests an increased concern with phonemic accuracy, coming about with the standardisation of spelling accompanying printing. Which sounds at odds with the chaos of Early Modern spelling; but Middle spelling was likely even more chaotic.