It's worth looking at the wiring diagram, but off the top of my head I can only come up with 3 reasons why the indicator would take a long time to turn off:
(1) there's a relay that isn't unlatching for some reason
(2) there's a capacitor in the circuit that is slow to discharge
(3) IDK what #3 is but as I was typing the first sentence I corrected 2 to 3, and now I'm committed to coming up with 3 reasons.
I would be surprised if there's a relay for the indicator, but there is a precedent for the instrument cluster to have a relay; the rear window defogger has a timed relay built in there somewhere, that's how the defogger turns off after 15 min. I can't think of why the indicator would need a relay though, nor can I figure out why there would be a cap in that circuit. I would have guessed it was wired in the same circuit as the high beams; whenever one is on the other is on. I should know this but I don't...do the headlights get power through a relay? If so it still doesn't explain it because then the high beams would stay on as well. So that's a lot of words to say IDK, but now I'm invested and I'm going to look at the diagram. brb.