I have my own situation with stuttering. I can speak normally, I will not attempt public speaking, but I can speak normally in normal situations.
I think that the cause of stuttering is physical and physiological, the parts involved are basic and the failure mechanism is simple, it happens inside the mouth, with the tongue, saliva and the chewing motion of the mouth. It’s not other parts of your body, your mind or the environment, they could have contributed worsening of the situation but they are not the main cause.
If observe your mouth, around the root of your tongue, even when not chewing, saliva would collect in the back inside the mouth. When you speak, you move your tongue and jaw to speak certain words, but you are subconsciously afraid that the movement would cause the saliva to flow out. Saliva is natural when eating and chewing food, but when you speak, the presence of saliva can interfere with movement of the tongue inside the mouth. You don't want to stretch out the tongue when speaking certain words for fearing that saliva would spill out, your jaw is afraid to move because it might cause saliva to spill out of the edge of the mouth.
If you keep the mouth closed, and it doesn’t even have to be very long, saliva would accumulate and collect inside. You can feel it. But just inside the lips, for some people, saliva might dry up and become sticky, and then at the moment when you open your mouth to speak, you feel that your lips are stuck together and causing the opening the most difficult when start speaking.
For me personally, saliva always accumulate on the right side inside my mouth, and the right side is the natural side where most movement occurs and is heavily involved when speaking or chewing food. When speaking, force is usually applied to the right side. Let’s call it the dominant side, maybe there are those who don’t have a dominant side.
To confirm this observation, lie down in bed, in my case of the right side being dominant side, lie down on the left side so that the right side inside your mouth is less interfered by the saliva accumulation and the movement is more free. Or when sitting, lean your face on your left palm with your left elbow on the table so the right side is elevated and saliva flows naturally to the left side through gravity. Experiment with speaking without the fear of saliva accumulating and interfering with the movement of the tongue, the jaw and the whole mouth.
In normal situations, before speaking, find a way to suck out the saliva collected inside your mouth, especially around the root of your tongue. If you still feel sticky inside, sip some water, rinse and swallow, for the moment when the tongue is free from saliva interference, speak normally, position the tongue, the jaw and lips in the normal fashion to speak, see if there is in improvement in your speech pattern when saliva is not interfering.
If you observe the effect and indeed there is improvement, then figure out a way ways reduce the interference because saliva is natural occurrence. When you speak, the motion is similar to the chewing motion when you eat, so saliva just naturally come out. Even when you are idle, saliva still occur by itself, you cannot stop it, you can only become aware of it and try to reduce its effect, and then re-learn the normal way of speaking.
Different people have different words they usually stumble upon. For example, some people stumble on the k sound like key or tr sound like strategic. If you say the words and observe the tongue movement, it’s likely that some people are afraid of completing the full movement because of the subconscious fear of saliva flowing out and innate need to control it, but that subconscious control mechanism would interfere with normal speech pattern.
For those people who don’t stutter, maybe their physical development became such that the physical configuration of the mouth, the tongue, jaw and lips, became natural at dealing with saliva accumulation and interference, or maybe there is no interference in normally developed configuration. But for people who stutter, you just have to be aware of the interference consciously, deal with it and adjustment the movement, and re-learn the normal or adjusted way of speaking.
When you see baby start speaking or just babbling, usually there is drooling, and it may be funny. But then they grow and naturally develop normal speech pattern, but some people may encounter difficulties during this development, the problems can definitely come from other physical areas and psychological nature, and show up on the face and the mouth. You just have to realize it and compensate for it.