You must keep in mind that ASL is not English produced on the hands. NOT is a form of negation. For example "I don't have time" could be signed as:
_____________neg
I NOT HAVE TIME.
"I don't have any children" However, would utilize the sign NONE
CHILDREN I HAVE NONE.
The text Linguistics of American Sign Language (Valli, Lucas) notes that NOT is not actually required in negative sentences and may be used for more emphasis.
You may be wondering how a person will show negation in absence of a specific sign. This is where NMM's (non-manual markers) come into play. For negative sentences they are marked with the word 'neg' NMMs in this category can be made by shaking the head side-to-side, by frowning, 'wrinkling' the nose, having a raised upper lip, and squinting the brows.
so the phrase:
______neg
MAN HOME
means the man is not home. A NMM is enough to make the phrase negative.
There are various other forms of negation such as reversal of orientation. You can see this in certain signs such as don't-WANT, don't-LIKE, don't-KNOW, etc. It is important to note that reversal of orientation cannot be applied to every sign. For example, it cannot be applied to a sign like HAVE. However, NONE would work well with HAVE if you wanted to say 'don't have...'.
I am by no means an expert but using DON'T would definitely seem a lot more English to me than using NOT and NONE.
See Dr. Bill's note on the sign DON'T (link below). I have also heard that this sign is used more as an indication to stop doing something (an action) but I cannot confirm this right now. Like I said, I am no expert but know that NOT can definitely be used to show negation taking the place of don't.
http://www.lifeprint.com/asl101/pages-signs/d/dont.htm