Does "stay dry" here mean simply not wet?
"Stay dry", because of the use of "stay", has the nuance of meaning continuing to be dry or continuing to not be wet. So we can infer that the cat is dry and wants to continue to be dry. This is a common use of the word "stay" to mean continue in a state or situation. Ex: Are you going to stay in bed all day? Ex: If I stay quiet they won't know I'm here.
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