It sounds like you need to do a little more research. Unfortunately, since the Paddling.com website changed its format a year or so ago, it is more difficult to find the basic “introduction to kayaking” and “how to choose a kayak” articles that used to be accessible here.
One problem is that you are expecting too many conflicting capabilities from your first boat. “Stability” is often a concern of beginning kayakers, and while it is true that a nervous beginner often feels that a kayak is “tippy” or “unstable”, that is more due to the inexperience of the paddler and not the fault of the boat. It does not take long for that “tippy” feeling to go away once you get used to being in the kayak and trusting it. Short, fat and wide kayaks do tend to be more stable feeling at first since they are a broad platform. But being a wide short boat has drawbacks if you are doing anything but just sitting in it on a calm pond.
You also express that you want something that is maneuverable and not slow – but most wide short boats are slow and many are barge-like and don’t track well. The short boats (under 12’) generally are wider than longer boats – they have to be to provide the volume of displaced water that will support the average adult. But you are smaller than the “average adult” which means that you will not fit in most kayaks very well. If a kayak is too big for you, you will not have good contact with the boat to control it, will have to reach farther over the width of the beam to paddle it, will sit too high in the water because of lighter weight, and also may be sitting too low inside due to your 5’ 1" height – also short boats are harder for shorter people to load on the roof of a car. Longer boats are easier to load. A longer boat will tend to be narrower and thus the same weight or even less than many shorter boats.
you also have to consider if you want an open sit-on-top kayak or a closed deck sit-inside. There are advantages to both. Sit-on-tops are usually preferred for fishing and for carrying a dog. BUT, they are a wet ride and not so good if you plan on being out in chillier weather or in colder or rough water. They also tend to be heavier and don’t have the protected storage hatches for gear that sit inside kayaks have if you are planning to do overnight trips in the future. You can stay warmer and dryer in a sit inside kayak and they tend to be faster and lighter per foot of length.
And in kayaks, you get what you pay for. There are options under $1000 new, but we usually recommend beginners try to start with a used kayak because you not only get a better bang for the buck, but until you have paddled for a while you are not going to know exactly what you want in performance or features. It’s pretty common for people decide within the first season or two of paddling that they want to change to another boat with different capabilities.
Do you live somewhere that has one or more dedicated kayak specialty dealers? Not the “big box” chain sporting goods stores , but an outfitter store (usually independent) that not only sells kayaks but also offers on-the-water demo days and instructions in paddling technique and safety, Another option is to see if there is a kayaking group (MeetUp often has them in major metro areas) where you can join outings – often there are folks who will share “loaner” kayaks with beginners to allow them to participate and this is another way to get some experience with various styles of kayak and get a better feel for what could work for you. One drawback, both for borrowing kayaks or finding a used ( is that you are not liable to have a chance to paddle many boats that are best for someone as petite as you are.
There are models that are scaled to smaller adults. You might even fit in some of the youth sized kayaks though many of them have an optimal upper weight range of 130 pounds and the cockpits might be uncomfortably snug depending on your proportions. The Wilderness Systems Tsunami 120 could work for you – they are no longer making that size of the Tsunami model but they were popular and fairly common used. Both the Perception Carolina 120 and the similar but less well outfitted Conduit 13 might also be suitable, and both are under $1000 new as well as fairly common used.
One new model, if you could find one to check out, is the Venture Islay 14 sit on top, which has some features that make it more comparable in handling to a sit inside kayak. I see at least one dealer selling it for $765.