"in states without a lieutenant governor, who is usually next in succession to the governor's office? "

The correct answer is the leader of the state senate.

They are people who are basically second in command to the official governor. It's something like a vice president is to a president. If the governor dies then the lieutenant governor takes their position, and if the lieutenant governor dies, then the leader of the senate for that state takes it. This is common and is found in almost all states.