God "oversees" natural disasters. That is, the Bible teaches that we live in a fallen creation that is "groaning" for redemption, so God allows natural disasters to happen.
He oversees them in unseen ways to where He works all things together to bring His ultimate will to pass here on earth. It's like you hear stories about people who were miraculously spared while their neighbors were not. Or you hear stories about how people who suffered in a disaster came to Christ or how God used His people to come in and witness for Christ here on earth after a disaster occurred.
As to imputing responsibility to God, the thing is that the responsibility lies on the shoulders of Adam and we as his descendants have to suffer for what our ancient ancestor did when he disobeyed God and brought God's curse upon creation. It's just like how a whole nation of people have to suffer when their leader makes a wrong decision even though each individual in the nation didn't have anything to say about it.
Romans 8:18 For I consider that the sufferings of this present time are not worthy to be compared with the glory which shall be revealed in us. 19 For the earnest expectation of the creation eagerly waits for the revealing of the sons of God. 20 For the creation was subjected to futility, not willingly, but because of Him who subjected it in hope; 21 because the creation itself also will be delivered from the bondage of corruption into the glorious liberty of the children of God. 22 For we know that the whole creation groans and labors with birth pangs together until now.
Genesis 3:17 Then to Adam He said, "Because you have heeded the voice of your wife, and have eaten from the tree of which I commanded you, saying, 'You shall not eat of it': "Cursed is the ground for your sake; In toil you shall eat of it All the days of your life. 18 Both thorns and thistles it shall bring forth for you, And you shall eat the herb of the field. 19 In the sweat of your face you shall eat bread Till you return to the ground, For out of it you were taken; For dust you are, And to dust you shall return."