is there a way to add another primary button to a native record? they want a button like the Save button that says Save As Draft (which will just check a custom Draft checkbox)

You can def script it but a workflow may be faster here to add a button. Form level button is also an option to get going.

cool will try that. Will the button come out as blue like the save button?

no it comes out the other style. Only the submit one gets the blue. You'd have to do some jquery dark arts to mess w/ the styling to get it to be the same