It appears that the buyer is not familiar with all of the ins and outs of EBay. Maybe be more an
amazon buyer where there really is not any communication.
If your listing is accurate and the handling time is posted, it should be fine. The buyer has responded even if not how you prefer.
If she has paid, send a message thanking them for payment and simply stating that the item will be shipped out on XX/XX/XX. If they have questions, to feel free to contact you and that you hope they enjoy the xxxxxxxxx.
I have found that some of my best buyers, especially on Bonz, are brand new.
They google what they are looking for and the listing pops up; they sign up and buy.
I understand your discomfort but this is how EBay works. You are not supposed to discriminate against a new member. You have the right to cancel, block etc but you are opening yourself up to negative feedback and NPS, I believe.
Good luck.