I was wondering whether the cause of this problem was an inconsistency in the way the height information is provided in the DEM and the way SNAP interprets it (a matter of including the geoid height or not). See the following thread for more details:
http://forum.step.esa.int/t/location-coordinates-shift-after-processing-ascending-and-descending-passes/
If it is just an issue with inconsistent use of height information, a sequence of images with unique orbit track should not show jumpiness. The geocoding of each one of the images will be equally inaccurate.
If the jumpiness is still there, then there is another unrelated issue.
Yes indeed the coast line moves as well. AFAIK most of the S1 images (could be all by now?) delivered via the Sentinel Hub have been processed with Restituted Orbits, which are already very accurate. If you need more accuracy, you can update the product with the Precise Orbits (Apply Orbit File in SNAP), but you will need to wait 20 days after image acquisition for the precise orbits to be published.