I would suggest that before you do anything you go to Georgina Terry's web page and watch every video and read every document she has about bike fit and what you are up against as a small framed female. She has made her living by dealing with the problems you face with bike fit. She explains the design constraints and the compromises that have to be made in order to build a bike for someone who is smaller. You need to know what these are in order to buy a bike that fits you and to cope with what sales people will tell you in stores.
She has good information about what bicycle companies do when they make their build choices and how it effects the bike and the rider.http://georgenaterry.com/bicycle-frame-design/
The information there will give you the knowledge that most bicycle shops will not explain or perhaps even know about fitting a bike to someone your size.
I understand about staying within a budget, but remember it is cheaper to buy once and have something work for you than to buy twice when the first thing you buy doesn't fit. Your main concern has to be the fit of the bike.