Lucien Piccard Da Vinci Automatique repair?

Can anyone recommend a good repair shop for this watch? I'm told it has a
"special" A.Shield 1800 mechanism with an offset something or other.

The watch is a "Lucien Piccard Da Vinci Automatique", 14k gold, roman
numerals at the 12, 3, 6 and 9 position and single bars at the remaining
positions. It displays day and date, vertically aligned at the 6 position.
At one time, the crown was a blue crystal (sapphire?) set in gold.

It was at my local watch repair center for about 2 years waiting on a part,
however when the part was finally located, it didn't help. Apparently,
there is a shaft of some sort that is worn out and will no longer hold a
gear that moves the hands. I think that's what the man said. The watch will
wind when I shake it but won't hold time. When I tried to wind it manually,
the crown and shaft(?) came out in my fingers.

I appreciate any suggestions/advice.

(yes, I've googled it)

Thanks

Dan
Cedar Park, Texas.