I see three issues that are all connected and explain the shoulder pain.
1. Throwing off the back foot--you are starting to pull the disc forward while the weight is still on the back foot. You can see this in how when you plant the front foot it is already opened to the target. The weight should be fully on the plant foot before the front swing starts. I did this for years and is hard on the knee and kills power.
2. Tipping--you lean back during the backswing (weight behind the back foot) and tip forward past the plant foot in the follow through. This feels powerful but robs distance and causes you to bend and twist your spine in bad ways. It will also make it hard to throw accurately consistently.
3. Collapsing and rounding (hugging yourself)--your upper arm is pushing into your right pec area when you pull the disc forward. As a result, all the power is being generated by your arm and there is a tremendous strain on your shoulder.
It's good that you are going to fix this after just a month--I had nearly your exact swing for 10 years and it is a lot harder to unlearn.
You need to learn to throw with the big muscles--legs and core/back. The best drill for this is to take a hammer, hold it dangling straight down and start swinging it back and forth like a pendulum, just a few inches at a time--but don't use your arm at all. Use just your feet to get the pendulum started back and forth. You should feel the weight move back and forth between your feet. Then swing the hammer slightly further on the same plane--still without any arm. As it goes beyond a bit further back, you will naturally turn your hips and shoulders back to keep the hammer going back. Don't force the hammer, just use it's weight and gradually add a little "pump" to it like your legs on a swing. Note how your body shifts slightly to resist the weight.
The feeling you should get from the hammer swing is that during the back swing the weight shifts to the back foot, then to the weight shifts to the plant foot, then the hammer starts coming forward. All of this happens without any input from the arms (other than gripping the hammer). I have literally spent hours doing this, focusing on making it smooth and effortless.
Next take a disc and start doing standstill throws. I start by swinging the disc back and forth like the hammer and getting the feeling I got from the hammer drill. Next, instead of swinging it back-and-forth like a pendulum, I swing it back like a pendulum, and forward like a normal throw a few times. Then I start throwing.
Doing the hammer drill and the standstill throws with the hammer swing feeling should get you throwing with the whole body, from the ground up, with weight on the plant foot.
