When the bible speaks of God sending an evil spirit, it is possible that the author means God allowed it to happen. Because God is in control of all things, even secondary causes, the biblical authors would often attribute to God direct causality even though they understood certain events to be a result of his permissive will.
With that said, it is not contrary to God’s good nature to allow evil when He wills to bring about a greater good. Consider that parents often permit certain evils to occur in the lives of their children even though they have the power to stop it, e.g., when a mother pours peroxide on the wound of her child and the father, who is watching, let’s her. The permission of evil, even when one has the power to stop it, doesn’t necessarily count against one’s goodness.
Here is an article that you might find helpful that deals with the topic of God allowing an evil spirit to torment Saul: gotquestions.org/evil-spirit-Saul.html
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