In the OHP, when I increase the weight above a certain amount, it sometimes feels as if a muscle in my shoulder is misaligned, which causes pain, and keeps the muscle from contracting.
But if I stretch my arms over my head and pull back, similar to seen in the drawing below, it usually seems to reposition the shoulder muscle, enabling me to resume the pressing motion.



If that description is understandable, does anyone here know what is the problem, and what is necessary to fix it?

Well number one would be making sure you are recovering properly

Tied for number one is your form, not like i know what the fuck i am talking about or can diagnose you without video or something but possibly too narrow of a grip/ elbows too far forward combined with a loose muscle.

Take a vid and the guys here can see if it's a form issue.

I am not aware of any issues with recovery.
My grip is a little more than shoulder width.
I don't have the capability to make a video.

I have to do a few behind the neck pressing prior to my strict presses to make my shoulders happy.  If I don't I get some impingement.  A few BTN presses loosens it up perfectly.

Decided to give it a try during Thursday's workout. Used just the bar, which weighs virtually nothing. Darned if it didn't cause pain in the other shoulder...

Shoulders are all about balance, if the tone and proportion of the muscles making up the shoulder are out of balance, their will be problems with the joint.

Gentle stretching, working the weakest movements and oddest articulations of the joint, with high reps of very light weights, until they begin to feel heavy and never going until it becomes painful during these odd articulations will bring the shoulders into balance.

Circulation in the muscles of the shoulders will be noticeable after balance conditioning but nothing like what you will see in the arms by simply balancing the muscles of the shoulder.

Shoulder will recondition themselves quickly but because of the small size of many of the muscles and the limitations of the joints range of motion, they are very easy to injure. If it pinches, pops or feels like it it's sprained don't do anything heavy with it until you have taken the time to balance it with stretching and very light weights. If it is simply tight feeling, gently work right through it with very light weights to start. 

A few days of balance conditioning and you will be back at the heavy work with improved results. Tweak your form on moves, a simple twist of the wrist if your using free weights will prevent injury.

Dangle, hang from a pull up bar, you don't need to do the pull ups to get a beneficial stretch but if you can then do a few of everything form you can muster carefully while keeping balance of muscles strained in mind.

Whether you are healing and injury or recuperating after heavy shoulder work, learn to sleep a few hours with you arms outstretched, practically straight not bent at the elbow, over you head. Most won't be able to comfortably intertwined their fingers to lock the position nor allow their arms to lay flat.

Use a couple of pillows positioned off each side of your head to rest you elbows on for the first few nights and gradually work your way too less and less support until they will lay flat. Use a loop of rope six or so inches in diameter to hold with your fingers to keep your hands together until the shoulders have relaxed enough to allow you to comfortably intertwine your fingers.

These things will banish most persistent shoulder issues effectively and prevent new issues from arising and also relaxes your chest and makes for easy deeper breathing.

Maintain proper hydration and educate yourself on joint and connective tissue nutritional needs and study the dietary cause if inflammation so you may avoid them. It is easy enough to get what you need from a healthy diet but keep in mind many nutrients are dependent on others for absorption.