There are many, many different rein holds for the double bridle. The one shown in the OP's picture is correct, but not ideal for her riding level.
What is the point of riding in the double? I'm assuming you're trying for Juniors? In any capacity, to save the horse's mouth while learning with two reins, attach both reins to your snaffle bridle and hold them that you snaffle rein is between your ring and pinky fingers and the curb rein is between your middle and ring finger. This lessens the effect of the curb marginally and is the most common rein hold for juniors and amatuers who do not yet have an understanding of the double.
As far as your position - check your saddle fit. I think the cantle needs to be raised to allow you to sit more towards the front of the saddle and your stirrup lengthened gradually by two holes in order to allow your leg to relax downward around the barrel and fall in the correct ear-shoulder-hip-heel line. You also need to greatly expand your chest, push your shoulders back and look up with a balanced head. The horse follows your head and if you're looking down - he is on the forehand.
Finally, to correct your kinked wrists, you need to drive your shoulders and elbows into your hips and keep the whole upper arm beside your upper body. The soft contact comes from a soft elbow and wrist and feeling hands. A kinked wrist is not soft. Keep your wrists straight, with the pointer fingers more towards the front of the horse than the pinkies, and keep your fingers soft. Shorten your reins until you have contact - don't pull back until you have contact.