Are there any examples of any English words that are both synonyms and homonyms of each other?
I would guess that over time one would become considered an alternate spelling and die out, so perhaps were there ever such words?
You may find this page of homonyms interesting. Some homonyms:
Air – oxygen / a lilting tune
Fair – equitable / beautiful
Ring – a band on a finger / something circular in shape
If you said, The air is fair, then it could be a weather report or a comment on a song you just heard. The word air in current parlance is the atmosphere, but in archaic language it means a song-like vocal or instrumental composition. The words air and air are homophones and homographs. While they are not exact synonyms, they do have the archaic / modern contrast that you are seeking.
Also consider the ring on your finger and a boxing ring. The former is round, by necessity, and the latter was round before it became the current square. These are homophones and perhaps close enough to be considered synonyms. And this is admittedly a stretch: Once again, the boxer took on the burden of the ring. Is he fighting again? Or getting married again?