I concur, as usual, with the good Doctor T.
As to washing up referring to washing oneself, as opposed to washing dishes, it can mean either in US dialect. The reference is determined by context. If you’re called to the dinner table and say, “I’m going to wash up,” it would be taken as meaning you were going to wash your hands and will be at the table in a few minutes. If you were getting up from the table after the meal and say, “I’m going to wash up,” it would be taken as meaning you are going to clean the dishes.
(Incredibly OED’s first cite for washing-up as a noun with the sense “table utensils awaiting washing up” is 1972! They really couldn’t find anything earlier?)
Undoubtedly due to the date of the entry, which was written in the 1920s, with some additional senses added with the 1970s supplements. I’m sure when the editors get around to the full revision, with the internet now at their disposal, antedates will be easily found.