We are to believe God's Word, not understand the mechanics of how God sovereignly rules His creation. All I can tell anyone is that the Bible teaches that God is absolutely sovereign and that the Bible teaches that man is absolutely responsible.
Where the problem exists seems to be in the definition of “sovereignty”.
Sovereign refers to the independent authority and unquestionable rulership of a king.
Historically, it never means that a king controls every atom, thought, and action happening in the kingdom. The king rules by issuing and enforcing decrees that the kingdom subjects are to obey, or they will suffer punishment. No king would issue a decree, see it violated by subjects, then claim he willed the violation to happen.
God’s hardening of pharaoh’s heart was accomplished by the miracle punishments inflicted upon pharaoh and Egypt. Pharaoh responded to these bad events by becoming more stubborn.
It cannot be affirmed that all happenings are willed by God to happen. But all happenings are under His rulership, meaning that rewards or punishments imposed by God will result from each thing that happens.
A parent may see its child copying a famous poem from a poetry book, then taking it to school to present it as their own original composition. The parent does not cause the plagiarism to happen, but merely permits it, without interfering or preventing it. The parent knows the child will be publicly humiliated by the teacher in class when the plagiarism is instantly detected.
The parent decided that letting the child’s plagiarism happen, and get punished by the teacher in front of the class, will make a better impression than if the parent interfered.
The parent’s sovereignty over the child does not mean the parent controls every movement of the child, or causes all the child’s actions.