When short on time, and if looking for ways to make your hip and shoulder stretches more challenging, one option is to combine the two.
Using twisting as an example, you can combine a simple seated twist with different shoulder stretching options shown below.
To improve the twistability, if twisting to the left press down with the right foot so that the right side gluteus maximus activates. Then try to activate the left side latissimus dorsai.
These two muscles are connected via the rear layer of the thoracolumbar fascia and you may find that activating them helps to deepen your spinal twist and make it more comfortable.
It also helps to tie together the opposite side shoulder and hip.
A very simple standing hip and shoulder stretch is standing side bend. Bending the spine to the right and pushing your hips to the left you can reach both hands past the head and grab the either the right or left wrist.
You can then stretch the left shoulder by pulling down with the right arm. Notice the difference in sensation when stretching the left shoulder with elbow bent versus straight.
The hip stretch in this pose comes from pushing the pelvis to the side. Assuming the hips are being pushed to the left, you then stretch the abductors of the left leg and the adductors of the right leg.
Both of these are forward bends for the hips. Thus they stretch the hamstrings. They also stretch the shoulders. In the wide leg forward bend the hands are clasped and pulled rearwards and over the head (dwikonasana) while in the triangle forward bend the hands are in prayer behind the back (reverse prayer).
Eagle pose (above left) is also a combined hip and shoulder stretch. It stretches the top leg and both arms in internal rotation. If you reach your shoulders forwards in the eagle arm position, you may also find that it stretches the rhomboids.
Another balancing on one foot hip and shoulder stretch is dancer or "balancing bow pose" (above right).
Start off on one foot and grab the other foot behind you. Bend the spine backwards first then reach the knee and foot back to stretch the hip flexors of that leg. You'll also be stretching the front of the shoulder of the grabbing arm.
If you can create enough stability in upright pigeon pose (press down with the front knee) you can release the hands for any of the previously mentioned shoulder stretches. The hip stretch in this pose can work on the outer hip muscles of the front leg or the hip flexors of the rear leg. To work on the hip flexors either press down through the back foot or focus on lifting the rear thigh so that hamstrings and gluteus maximus both activate.
You could also bend the back knee and stretch the same side shoulder and hip flexor by pulling back on the foot.
You could also lay on your belly and stretch both arms and both sets of hip flexors at the same time in bow pose.
Some binding poses which can be used to stretch the hips and shoulder simultaneously include ardha Matsyendrasana (top left), Bharad Vajasana (top right) and compass pose (bottom right). (You can warm up to compass pose with compass prep at the bottom left).)
One other simple combined hip and shoulder stretch is double pigeon with hands clasped and pressing upwards.
As a yoga teacher, I'm constantly exploring new exercises, new ways of doing yoga poses.
There is no single "right way" of doing a yoga pose. Instead, there are options. And the better you are at "feeling" your body, the better you can get at choosing the right option for your body as it is now.
For any technique, the point of practice is to learn feel it and to control it, so that it can be used without thinking about how to use it.
And that is more or less the approach taken in all of my ebooks and videos. They help you to feel your body and control it so that you can work towards using it effectively in anything that you do.