What you are describing is called strabismus. Strabismus means a misalignment of the eyes. There are certain types of strabismus where the eyes move up, out, in, or down and it sounds as though yours is the type where your eyes move out. This is called exotropia. The fact that you do not have stereovision, or the ability to see things in 3D as you were told is a sign that this was a problem since your early childhood. When the brain does not hold the eyes together sufficiently during our formative years, this ability to see things in stereo does not develop and therefore is lacking the rest of our lives. You still are able to distinguish where objects are in space because you've learned monocular clues (shadows, angles, sizes of objects relative to other objects, etc).
The reason this exotropia happens more when you are staring at something is because the brain tends to relax the control of the eyes when you are not intently moving them and this relaxation causes your natural tendency for the eyes to move out to become more pronounced. This is also why this happens when you are more tired. The eye exercises you have been doing are called pencil push-ups and help strengthen your ability to keep your eyes together. Continuing to do these things is a good idea as this is one of the very few things that can help stave off worsening of this misalignment. Also, if you have any nearsightedness at all (ie myopia), even if you don't currently wear glasses, wearing nearsighted or myopic glasses can help pull the eyes together as well.
If, despite these two measures, the eyes continue to get worse with respect to this problem, the only recourse is surgery to realign the muscles that move the eyes. While it's still major surgery, it's a relatively easy thing for the eyes to recover from and involves very little pain or discomfort.
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My opinion is solely informative and does not constitute a formal medical opinion or recommendation. For a formal medical opinion and/or recommendation you must see an eye doctor.