Your mare is cantering heavily on her forehand, which makes for a terribly uncomfortable ride for both horse and rider. Being on the forehand means that she is essentially in 'front wheel drive' rather than 'real wheel drive'. We want a horse to be in rear wheel drive, because this allows the horse to be far more balanced, carrying more weight on the it's hind legs. Thus your 'steering' becomes easier as the shoulders are lighter, and the canter becomes more uphill, allowing you to sit far more easily.
Basically you are being a passenger in your riding rather than a driver. I'm assuming the horses youve cantered previously were school horses, which are generally in autopilot mode and maintain a steady, relatively balanced canter for a beginner to learn on. Your lease horse is probably not so well educated as a school horse, and as a result you are struggling to be the 'boss' when in the saddle.
You are doing yourself a disfavour by gripping with your legs and bum. As soon as your bum tightens, you're guaranteed to fly out the saddle. It's physically impossible to follow the canter motion with a tight bum and gripping knees/thighs.
I suggest you have a few more lessons on school horses, on the lunge, with no stirrups and reins, and preferably with your eyes closed. Sounds scary, but it will give you a really good feel of how your body should be moving to stay with the horse's motion, and also help you to start developing some skills to become the driver. By gripping yourself, you are just putting your horse further off balance, so she will go further onto the forehand, which will make you grip even more. It's not the horse's job to balance, that's YOUR job.
When she goes to rush in canter, take a big breath in, and slowly let it out through your nose. Just relax, go floppy. Allow your body to be taken with the canter motion. When you can relax, allow your legs to just hang, THEN you can start taking control. Hold your core muscles and feel that if you brace them slightly, your horse will slow down a little. This is the start of becoming a driver, learning to use your seat to control the horse.
As I said above, you can only control the horse when you have balance yourself, so take some more lessons on the lunge, and get a really good feel of how to sit into the canter and drive it, rather than just perching on top and hoping for the best.