I 2nd the idea of more turnout time. Horses can live 24/7 outdoors and be healthier and happier. I also 2nd the idea of a permanent companion. Goats work well, as will a pony or a mini.
If you intend to do dressage with this horse, her sensitivity will long term be a blessing, as it will take only a very minor shift of weight or balance on your part to tell her precisely what you want her to do. However, you will need to ride this horse with a very relaxed body, and that won’t happen until you can feel confident & relaxed on her back. Horses are incredibly good at reading body language, and minute changes and shifts in rider weight. Which means you that to be successful with this horse, you will have to be able to relax on her back, which is going to be mind over matter for a while. The best suggestion I can give you is to teach your body to relax on command off the horse first. This is something you can do while watching TV, and pick a body part and relax it. Fingers are a good place to start, then arms, then shoulder muscles, then feet, the legs, and so on, until you learn how to have good control over your body. It might also help to have a good dressage rider who knows how to ride relaxed ride the horse for a while to get the horse used to the idea of a relaxed rider so she can feel confident enough to relax.
The other problem with a rider being unable to relax muscles at will while riding is that horses, being herd animals with a head horse, from whom they take warning of possible attacks, tend to regard their human as a replacement head horse. Tension in a rider’s body is often perceived by the horse as the human/head horse stand-in being tense because they have spotted a predator or active danger, which is even scarier for the horse being ridden if they can not immediately identify the reason their human/head horse has tensed, which they cannot do as there is no threat or danger, just a tense rider who is tense because they expect the horse to do something dangerous & threatening every moment they are on the horse’s back.
This is not the type of problem that has an easy solution, and this may simply not be the right horse for you, or perhaps the wrong rider for this horse. There are people I try to avoid because even on a good day that person will rub me the wrong way, usually without intentionally trying to do so. Some horse/human relationships have a similar problem. This done make you a less than good rider, and if you see it as more a personality conflict than bad intent on the part of you or the horse, it might make any long term decision easier.