I'm 5 ft 4in so I have some idea what you're looking for. From my own personal experience the smallest decent folder out there is the Dahon Eezz d3. They're recommend for 5ft 7in and below. It maybe a little more than you want to spend but it's close to where your looking at. Also it very light weight and I can carry it up stairs very easily. It's not the greatest folding bike as far as folding but it works. When I'm using it everyday I don't fold it. It has a very short wheelbase, which is not great for riding but unfolded it doesn't take up much room. If carrying weight and size is a factor the Dahon eezz d3 is the way to go. It doesn't compare to a Bike Friday or a Brompton but for your purposes it would work. Your not going to find those bikes in your price range. One other bike is the Downtube Mini but I don't have experience with it but others seem to like it.
If you live in or near a large city Craigslist maybe the best answer. Go to bikes and search Dahon or folding bike. There's always someone selling a decent folder. The truth being that almost all folding bikes are good for short people. As someone who rides an xs frame I can say that it's a pleasure not to have the top tube right in my crotch.
One last idea is buying an adult scooter like a Micro Black. You'll have no issues with height and they fold down very small and are light weight. Also the Xootr is nice and the price is right.
Last edited by Pahana; 04-23-20 at 10:39 AM.