Can someone the difference between Моє and Своє ?
"Своє " is like "own", so "my own" here. "Моє " means "my". Both are the same here. because in the translation "own" is deleted, because it isn't used in English.
My instinct is to say Я шукаю своє місце. I would like to hear from a native Ukrainian speaker about this.
You are right, Я шукаю своє місце would be the best translation. Свій/своє is not covered in the current tree, but it is a good candidate for tree 2.0.