The hairspring needs to be adjusted at the stud, and then along the first coil past the regulator to be concentric with the balance center. It's way off, this is what's giving you your beat error change. You can probably see the spring body shifting as you move the regulator now. For the slow rate, check that the regulator pin is straight and close to the "boot", there should only be about 1 hairspring thickness of freedom (or less in many cases).
To adjust the spring from the stud, it would be best to remove the hairspring from the balance, install it in the overturned cock, and adjust. When you are happy with the curve that the regulator follows, you will most certainly need to make a new adjustment at the end of the regulating curve to center the spring. The collet hole will be directly over the balance jewel when it's good.
In the pic you can see a circle concentric with the balance center, and an approximation of the bends necessary at the stud, and where to make the additional centering bends at the red arrow.