OP, I’d urge you to reconsider framing this whole thing as an “extrovert vs introvert” battle. Your boss is demonstrating poor boundaries and disrespectful and inappropriate behavior, and that has nothing to do with whether she’s an I or E on the Myers-Briggs.
Whether or not you go to HR depends a lot on your company culture. Either way, you need to be documenting specific comments and specific behaviors that are inappropriate first. Every time she asks you an uncomfortable question, especially if it relates to sex, write down what she asked, how you responded (that you declined to answer and asked her to stop asking personal questions), who else was present, and the date/time. Keep this in a personal account, not company. Do NOT go to HR without documentation.
We don’t know how big your company is, but odds are if she’s a middle manager she’s got people above her already who know she’s a gossip and hate that. If you have any relationship at all with her supervisor, it’s generally viewed as following the chain of command if you bring concerns like this directly to them, as well.
Like I said, whether you should go to HR will depend a lot on your company culture. In all the jobs I’ve worked, I’ve had HR departments that would’ve taken your complaint seriously and not allowed you to be retaliated against in that way. Btw, what you’re describing does sound like retaliation, which is totally a lawsuit you could pursue