"하늘에 구름이 꼈어."
Translation:There are clouds in the sky.
I'm no expert so take this with a grain of salt but they might be related through Hanja - the Hanja equivalent to 그늘 is 陰 which as you can see includes the character 云, an archaic word for 'cloud'. Considering how some radicals in Chinese characters influence how a word is pronounced, this might explain why they sound similar? As for 그림자, the Hanja doesn't seem to resemble either word, but I did find someone claiming that the 그ㄹ part originally meant 'sun' so maybe there's a connection there.