Good questions. Those two issues definitely indications of muscle tightness in your shoulders, as you suggest, and likely your arms and hands too. There are many lines of energy and force running through the shoulders and into the fingers and when they tighten they can cause numbness and tingling at any spot in that line.
You do need to be careful not to push it too much or for too long, but mild tingling and numbness can be ok. In fact, it may be necessary, at least for now, in order to stretch the muscles that are tight. When you think about it, if a tight muscle is impinging on a nerve or blood vessel, when you stretch it there’s a good chance it will impinge a little more before it releases.
So experiment with the angle of stretch, intensity and length of hold until you find something that causes less of tingling and numbness and more of the release. And most important of all, do these stretches A LOT. And doubly most important, figure out what you’re doing (posture or stress related) that is causing the tension in the shoulders and try to work with that.