The owe you a free service and repair on this, they sold you a watch with damaged internals. The should be able to send it to Patek and take care of everything for you.
On my 3941, I noticed that the hand on the date subdial was well-centered on the dial marker at the end of the month, but that the alignment has gradually deviated over the past two weeks. I used the pusher to adjust the date to test the alignment through...