Should I use negative affixes on nouns, as opposed to verbs? I don't know of any natlangs which do this, but, conceptually, I think it should make sense in most cases.
Are thinking of having the nominal negation replacing clausal/verbal negation? That would be a bit weird, and I don't know whether it's attested in natlangs. But if you are thinking of having nominal negation together with verbal/clausal negation? That's extremely common - in fact, I think Wateu is rather unusual in not having any kind of nominal negation. (To say something like "I love nobody", you'll have to say something like "There does not exist any person that I love".)
Once you include nominal negation, there are a few things you need to decide about. How should the nominal negation interact with verbal/clausal negation? Especially, should you allow for double negation, or even make it obligatory? Should the nominal negation affix or word be the same as the verbal/clausal negator?
One further option exists. Negative affixes on nouns may be a kind of verbal/clausal negation (=their scope is the whole clause), only that they are attached to the nouns rather than the verb.
I was planning on having nominal negation only- no verbal negations, except in the occasional verb which is a "negated" version of another verb (I can't think of a good example, but something like English "run" versus "crawl"). I considered having clausal negation by the noun affixes, and I'm still willing to, but I think there could be some semantic fun in nominal negation.
Yes, you can try
German uses the nominal negation much more than English "Ich bin kein Mann" 'I'm not a man.'
If you have a scope of the negation, nominal negation is easy to attach to it. But if you have a just a (definite) subject: "The man (that doesn't exist) sleaps/The man doesn't sleap" are not the same. Of course you can use it in the form of your lang and ignore the semantic oddity.
From your example, consider "Not I sleep," this would be one way to negate the noun, showing that "I" did not sleep, while implying that other people do sleep; which is roughly equivalent to saying "I didn't sleep;" if asked "What did you do?", you would respond either "Not-me slept," implying that sleeping was an action which others were doing when you were not, or "I was awake", the positive equivalent of "I didn't sleep". If, then, you needed to say "Nobody doesn't sleep", you would have to either convert it to "Everybody sleeps", or say something like "Nobody stays awake." Does that make sense, or is there still some semantic quirks I've left uncovered?