If the unit that I have chosen to copy with Shady Spectacles die, what happen to the unit that Shady Spectacles attached to?

Yes — nothing happens to it. The equipped unit stays a copy of the unit you chose; the chosen unit dying doesn't detach Shady Spectacles or end the copy.

The printed duration is "for as long as this is attached to it" — the "it" is the equipped unit, so the copy effect lasts only as long as the Spectacles stay attached to that unit. It ends when the Spectacles detach or the equipped unit leaves play (the equipped unit dying kills the attachment, FAQ #12669), not when the copied unit does. There's also no re-choosing: you pick the unit once, as the Spectacles are attached.

So the equipped unit keeps using the chosen unit's copyable (printed) traits — name, type, tags, cost, domain, rules text, printed Might — and keeps everything of its own: buffs, statuses like Temporary or empowered, and any other gear attached to it (Rules 446.2, 479.1.b.1, 479.4).

One caveat the rules don't state outright: if the unit you chose was itself only a copy of something else (another Spectacles, a Mirror Image token, etc.), the copy effect's target is now a card in a new zone and its layer alterations are gone (Rule 124.1), so the traits carried over would be that card's printed ones. That's my interpretation — no official FAQ covers a copy source leaving play.