I think she’s scared and probably doesn’t trust you (since you only just got her).
I have a 26 year old Arabian x Quarter Horse large pony gelding that I’ve owned for 10 years and came from an abusive and neglectful situation. I don’t know if his personality would have been drastically different if he had been raised correctly, but he is very much a “pleaser” and even now is still fearful of potential punishment. He has excellent ground manners and lunges very well, but is typically at least a little tense. Like you were advised for this pony, I do things slowly and carefully around him so he doesn’t think something bad is coming (whereas my young TB doesn’t care if you throw his blanket on quickly or act a little silly with him and will stand there unphased). But like your mare, he won’t go run around turnout to get the extra energy out (either in an arena or a field)…he just stands there.
What worked for my guy was lots of positive handling on the ground to build trust. It doesn’t all have to be very structured training (maybe your mare already had great ground training anyway) but even just positive experiences doing daily things with you (grooming, taking on short walks in-hand, bathing, leading in and out of turnout, etc.) will help her build some trust and faith in you as her leader. Then when you go to get on her back, she won’t have as much to worry about. And once you get on, ride with a no-pressure kind of attitude. As in, “oh we’re just going to take a relaxed walk from here to there, a little scratch on the neck here and there, then let’s maybe just say whoaaa and stand here and scratch the neck a little bit, and go on and walk over here.” I would be conscious of how you’re gripping the reins, how fast you move your hands while sitting on her, how quietly you apply leg when you need it, and the progression of forward aids (be easy when you mount but keep a steady feel of your leg on her from the start so when you apply leg to go forward, it’s not suddenly hitting her sides instead of already being there and just making a little more pressure…also maybe start with some quiet easy voice aids). Give lots of neck scratches and rubs and quiet/low voiced reassurance. Keep your body relaxed and your attitude low pressure. Start out your rides with the goal just being to amble around at the walk (it may not start or feel like a slow amble, but don’t hop on with the goal of riding her on contact and asking her to correctly bend her body and so forth). For a while it may just be some quiet walking rides. Maybe eventually ask for a trot (and with this kind of horse, I wouldn’t worry about getting an immediate response yet, the quieter and slower you apply the aids, the less likely she’ll be to shoot forward) but just down the short side, then walk and try to amble again. Build up to more. Don’t think about jumping yet. You’ll feel her when she starts to trust you and can kind of breathe and go more consistently. But usually with this kind of horse (if she’s anything like the abused gelding I have), time, patience, and trust are your friends.
This is, of course, after you’ve checked saddle fit and checked that you have the right bit for her.
My very skittish gelding did end up being a lovely ride for me (not so much anyone else) and hasn’t tried to shoot forward from me in years. He no longer feels like a tense bundle of nerves when I ride, but I also am careful to kind of respect his personality and not be rough or playful or whatever with him. I never move my hands suddenly when I’m sitting on him, I am careful about how I swing my legs (when getting on/off or adjusting the stirrup length), I never raise my voice while sitting on him, and I give him quiet praises and a little scratch here and there when he does well to give him some confidence.