Haha, you gave me a funny picture in my head. If you told Paul to wash a weary face, he would probably be looking around for someone else's weary face to wash because you can't tell him to wash his own face that way;
a tells the listener that we are being generic. If you wanted him to wash his own weary face, you'd need a possessive determiner (your). So, yes, you seem correct.
When thinking about determiners (articles are a category of determiners), you always need to decide if the noun is count or non-count, and if it is count, if it is plural or singular as well as if it is generic of specific. With non-count nouns you need to decide if the noun is specific or generic. Once all of that has been decided, you still have lots of choices.(Go to Wikipedia's "Determiners" page to see the long list.)
If you want, send me an email and I'll send you a PDF graphic that I give to all my grammar students. Determiners are tiny, but powerful and confusing, little words. I'll send the file to anyone who wants it. Just send me your email.
~Miss Mandy