Reddit reviews Airgoesin 2 Stainless Steel Tonsil Stone Removal Picks Tonsillolith Premium Tool Oral Cleaner +1 Irrigation Syringe with a Case and a LED Tonsil Remover Tool

Dentist only messes with your teeth. Not your throat. I would say your PCP would be the most likely... I've never heard of it being done before but honestly if my doctor asked me to I'd grab a 50cc syringe I use for ear irrigations and shoot some water at them until I saw stones coming out, and then had you spit. To me that sounds kinda crazy and weird so I highly doubt a doctor would do this... but I could be wrong.

Like I said, I really do recommend a water pick. Or get a syringe of sorts. I know you mentioned having an issue with coordination, but it's a hell of a lot easier to maneuver around with these than it would with just your finger or a Q-tip.