The sentence you wrote is almost correct! Here's the sentence with a small grammar fix:
"Elephants stay cool by spraying water and fanning themselves with their big ears!"
Explanation:
• We add the word themselves after fanning to make it clear that elephants are fanning themselves.
• Everything else looks great!
So, your sentence is easy to understand and fun because it talks about how elephants use water and their big ears to stay cool.