I'm sorry if this is a duplicate question (I bet it is), I didn't really know what to search for.
猫を描く、犬を描かない。 (I draw cats, I don't draw dogs.)
How do you say that sentence without having to repeat 描かない? So it would be 'I draw cats, not dogs.'
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Depending on how attached you are to the original meaning, you could go with something like "Instead of dogs, I draw cats": 犬のかわりに猫をかく。
Or go into why it is you don't draw dogs: 犬が好きじゃなくて、猫を書く。
But I'm not sure you're going to find a way to retain the simplicity and precise meaning of your sentence and not repeat yourself.
First of all, I am sorry to say you have not yet understood correctly how Japanese verbs conjugate.
These are English sites for the explanation of Japanese verb conjugation forms.
Let me continue by saying that the sentence you originally wrote is incorrect.
猫を描く、犬を描きない。 (I draw cats, I don't draw dogs.)
it should be 猫を描き、犬は描かない。
I think you are not yet familiar with the six patterns Japanese verbs take.
In your first case, "I don't draw dogs" can not be translated to "犬を描きない”。
The basic verb 描く "draw", needs to be changed to "描か” for the 未然形．
The Japanese site, http://www.hello-school.net/harojapa006.htm explains:
種類 ( type ) - 基本形 - 語幹 - 未然形 - 連用形 - 終止形 - 連体形 - 仮定形 - 命令形
五段 ( type ) - 読む - 読 - ま/も - み/ん - む - む - め - め
the verb "描く” is a 五段活用 and conjugates as above.
Forming the ない (negation) takes the 未然形 ( future form ) as its stem, and thus it needs to be changed to 描かない。
My personal recommendation for you is, after you master these conjugations, you'd better go to your original question and consider how you could master Japanese systematically.
PS : The answer you are asking for, is, actually, 犬では無く、猫を描く。