Very Unlikely.
Definitely you need a third party repair place. Apple won't touch that
They will try an upgrade you before anything. But i would try other repair places.
A couple years ago I took my Classic into this location and thought I'd ended up on Antiques Roadshow the way the tech acted. He thanked me for bringing it in because he hadn't seen one "in ages". It was no longer supported then and he offered to get me set up with an iPhone. I passed.
Like Ibraheem said, the apple store most likely will send you aeay. There is however another store at the Greene called Gem City Digital. I highly recommend you go in and talk to them as they were a bunch of help when i had issues of my own. They do practicslly anytging mobile elctronics, desktop computers, tablets, etc. You may be able to call and get a quick quote too. They also sell used/ refurbished prosucts as well. Good luck!
I would first try a different pair of headphones and see it the problem persists. If it does, I would reach out to Apple directly as this is just a Google service and not the store itself.
Any product older than 5 years is not supported in Apple repair shops, you will be directed to third party repair shops.
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