Agree with others to rule out/treat pain first & foremost. Aside from saddle fit, could be other back issues, teeth, bit, ulcers, hooves...
Assuming it's 'only training', firstly you've given very little info, so we're left guessing at a lot. But first point I'd guess at is that she's in a new environment, being 'worked' by a new person & she may be generally nervous & stressed by that. I would have given her time to settle in & get to know you first.
Second point I'm guessing is that she's not very well trained generally & perhaps doesn't understand leg aids - perhaps putting legs on only frightens her or means go/go faster. I start with desensitising them to being touched/handled, then teach them how to *yield* to all kinds of pressure on the ground first. How have you gone about the 'ground work' with her & have you worked on her understanding how to softly yield & not react to pressure in various areas?
If when you're riding she's rushing around already, regardless of leg pressure, I would personally work on getting her *comfortable & relaxed* at a stand still & then walk first, before then working on her understanding of leg aids. I'm not exactly sure what you mean by 'milking' the reins, but if I want a horse to slow, I start with a very gentle 'ask', but then firm up if the horse doesn't listen.
Especially as you're wanting to teach her to respond to your legs, I'd be using one rein, rather than 2, to slow/turn, which will help her understand what you're after. Eg. starting at the standstill... after teaching her to yield from pressure on the ground & making sure she's relaxed about you sitting on her in the first place - I'd take up the slack on one rein, start asking her to yield to that & lightly put your leg on to 'back up' the rein cue. That way, you retain control & combining the leg cue with one she already understands well, she'll start to learn that legs mean more than forwards. Once she's doing this reliably, you can start using your seat/legs to turn her first & back up with reins if/when needed.