The thing after the Shabbos meal [or any bread meal, for that matter] is called mayim achronim. That is a different topic, not related to negel vasser after sleeping.
Basically, after you finish eating a bread meal, and are ready to commence the recital of Birkas Hamazon, you need to first wash your hands to get rid of the Sodomite salt, and so that you can bless G-d with clean hands [as well as for other reasons]. There are different opinions regarding how much to wash [the Gra says to wash until the wrist].