I suspect this was still a dialectic distinction by the time of the Norman conquest; I suspect the distinction survived much longer in more linguistically conservative regions like what is now Yorkshire (which was also more heavily influenced by the danish.
I believe they can still have distinct pronunciations in Icelandic.
I do recall that ð was voiced in the old norse that I was taught in college. It probably doesn't matter too much since you don't have an old englishman to speak to.
The presence of two different characters has something to do with the integration of Latin and runic alphabets, but the choice of which was used isn't clear to me.
The above variations are in a tiny subset of en_ and across a probably tiny piece of time. Nowadays we have phones and TV etc which I think probably works towards homogenising our vocalisations.
I suspect that Old English was a bit more complex than is generally thought, if care is not taken.
I remember realizing at one point that I had read everything in Old French in the university library, and thinking that that probably meant I had probably read everything surviving in Old French.
I recall my sister in law studied Occitan. Oh: https://en.wikipedia.org/wiki/Old_French - another rabbit hole to bookmark for spelunking.
Icelandic employs a 19th century reconstruction of the old norse alphabet noting that "Various old features, like ð, had actually not seen much use in the later centuries" - which corresponds to their merger into one digraph 'th' in English. The article also notes that the letter ð undergoes a devoicing depending on its surroundings.
[0] https://en.m.wikipedia.org/wiki/Icelandic_orthography#Histor...
p.266 > Of these additional letters þ and ð were borrowed from Old English. The rune þ was known already, but its use in manuscript came from England.
p.268 > þ in the oldest Icelandic manuscripts was used both for the voiceless sound of th in English thin and the voiced sound in then. About 1225 ð was introduced, and gradually þ came to be used only initially, and ð in other positions. þ then represented only the voiceless sound, while ð, except when following a voiceless consonant (rare, as ð then usually became t) was voiced, as in faðir, við.
This is where I got the impression that ð is voiced.
*: An Introduction to Old Norse, E.V. Gordon, Oxford Univ. Press.