See schmatic of that area below. I'm assuming what you're saying is you used the turn signals, they worked normally, but when you turned the switch off the amber dash light stayed on.
I think your flasher unit has failed. I don't recall exactly where it is, somewhere behind the dash. Usually it's an aluminum or plastic box or cylinder about 1" on each side, and might look just like an automotive relay. If you can get it to make it's "tink-tink" noise, perhaps by manipulating the switch, you might find it that way. If you can't find it let me know and I'll figure out where it is. If you can, swap it for an equivalent from your local Pep Boys or whatever, and see if that fixes it. They're pretty generic and cheap.
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