Obviously not Hector either, but absolutely do not give any woman your passwords to anything. Even the Subway discount site (or any other trivial thing).
Any woman that wants that much control over someone is a MONSTROUS red flag to me. Any woman that tried to get my information like that would be immediately put into FB position and nothing higher.
Also, any justification she gives I would automatically assume she herself does. "I want to make sure you're not cheating on me" probably means she's cheating and etc. Stay safe.