While I couldn't tell exactly when we started to use dear, in Old English, a common term of address was 'leof', which literally means 'dear'. This wasn't just reserved as a term of affection, but one of respect, like we might use 'sir' today.

Old English letters typically begin with a formulaic greeting in the 3rd person. A letter from Alfred to a bishop might begin, "Aelfred cyning greteth Aelfrice bisceope freondlice ond eadmodlice." That's "King Aflred friendlily and humbly greets Bischop Alfric." After the greeting, they switch to first and second person.

Here's the first few lines of a letter. Note that after the greeting, the author adresses the recipient as 'leof':

"Ælfric munuc gret æðelwærd ealdormann eadmodlice. Þu bæde me leof þæt ic sceolde ðe awendan of Lydene on Englisc þa boc Genesis..."

"Alfric monk greets alderman Athelward humbly. You bid me, dear, that I should translate for you from Latin into English the book Genesis..."