Im not sure what you mean when you say you feel it in your shoulders. Feel effort? Strain? Discomfort? It is very hard to describe a feeling. You are throwing in the 70s in the video (guesstimating). My son does this unconsciously where he slows down his delivery in games. He doesn't know he does it, but, he does to help with control. Its a bad habit. So, he can warm up in the pen throwing 83-86 and come in a game and be sitting 80-81.
You are a big guy so people expect velocity, but, there is a lot more to it than just size. You seem to have good control so you may have great success being a soft tosser with command. There are guys like this in all levels of college ball.
When I mentioned before a "one piece" delivery, I mean no hip/shoulder separation. The lower body doesn't do much and it is an all arm throw. If you go watch some youth baseball you will see pitcher after pitcher doing whatever they do to start their motion then when it comes time to move toward the target they swing their front leg open and slowly rotate toward the plate and throw all arm. Because most kids have weak arms you will see all of this movement results it a weak "rainbow" toss to the catcher. Older pitchers (HS age) may still throw all arm but have developed enough physical maturity or arm strength to generate decent velocity this way. Your mechanics are smooth but they are not explosive. At some point to get a more powerful delivery the body (weight, strength etc) has to move quickly to the target. Lantz Wheeler has a description calling it as the "center mass"....the thighs to the belly button. Once that momentum is going and the front foot plants that momentum is moved up the body into rotational force, the upper body rotates explosively to delivery the ball. When you hear people talk about hip shoulder separation this is an expression to describe the lower to upper body sequencing. The lower body moves quickly into foot plant (hips open to the target, the opening of the hips activated by the hip not swinging the foot open), the shoulders which are still closed then rotate around that firm plant leg and you get a more explosive delivery. You don't see to have an unquick arm, which is good. To get an explosive (which will usually translate to more powerful) delivery it has to start with the lower body. Some describe it as a hip lead or pushing off with the drive leg or there are many other descriptions. The hard part is getting the pitcher to feel what all those different descriptions are trying to say. Basically, if the body is moving to the target in an aggressive way, explosive and fast with proper sequence the arm will be forced to move faster. The hard part is teaching it.
If you can looking up Paul Nyman or Lantz Wheelers materials is not a bad idea. They both break down sequencing in a relatively easy to understand way. Understanding it is a lot easier than getting the body to do it in an effective way. Sometimes little improvements here can lead to velo jumps.