我想玩游戏用手机 so it can't be in this order?
No, I don't think so, it sounds odd that way.
What about “我想用我的手机玩游戏。”？
That works :)
what about ”我想玩手机游戏“ ？
How about 我想在手机打游戏
The English sounds stilted and artificial with "would like to" instead of "want to".