I found out that I needed motor mounts to correct this problem of the jerking. I replace all 4. I ordered them( ONLINE through ROCKAUTO) and I won't go into another retail store again for parts. You can't beat their prices.
I have the same problem with my 2009 Sonata. It started last year at the beginning of cold season here in the middle east. It disappeared when summer came and now it started again last month when winter began. It does the jerking when down shifting from 4th to 3rd gear. My temporary solution is to shift to manual gear when slowing down, then shift back to automatic when accelarating again.