"I do not draw."
Context. You could ask the same for the english "I don't draw". Draw what? Your sword? A picture? How are you to know from just listening? So even in English you have to give context.
But this exercise isn't really about teaching you how to tell random phrases that you'd use like puzzle pieces. But rather on the concept of building this kind of sentence.
Think of it as a practice of language melody and rhythm. If you understand and rehearsed the rhythm, you can take it apart, play around with it and ask your Japanese friends or your trusty Google translator if you a) have pronounced it well enough for Google to understand you and b) what Google thinks what you mean, when you take apart the elements and arrange them in stupid orders. 描きはかきません。 かくはかくません。 描は書くません。 えはかくません。 絵は書くません。 Is it really a far stretch to imagine all of them could mean something different? That is exactly where you need help.
Well, be honest, you didn't think finishing duolingo alone would make you fit for a random Japanese conversation. Hence the lack of kanji. You don't really need them here on duolingo. You must do the research beside of duolingo. The hiragana tells you how it's supposed to sound. The rest is for you to explore.
The kanji is how you tell. The first one means to write, not draw. Context is also important but in this case the kanji let's us know that one verb means to write and the other means to draw ie. a picture. If there was no kanji to help us make that distinction then yes, you would rely primarily on context.
PDBPH - you are assuming that a verb in another language has the same meaning and usage as its English counterpart. Its probably good practise not to assume things about other languages. In this case neither would be used to mean to draw a sword.