This is similar to what grosales said but a bit more drawn out with some fancy psychology thrown in:
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I have struggled with a similar problem and know how debilitating it can be. What I mostly offer here is a few questions to ask in hopes of getting honest answers for yourself. These tend to turn into much deeper issues than they seem at the surface. Not that deep issues are bad, but just take longer to work through. To encourage the full reading of this comment, I have broken into three parts. I should also note that I am not a certified psychologist, but will pretend to be one if the price is right.

PART 1 of 3

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I think the key is in what you said here: "The main problem I have is the pervasive feeling in the back of my mind whenever I spend time doing something that I could be doing something else." When you say, "I could be doing something else", I imagine its just not anything else. Instead it's something that internally drives you to become "completely engrossed in my work and won't sleep, eat, etc. until it's finished".

What is it about this other stuff that seems so appealing? Because it can be different for everyone, let's just say it's "stuff that makes me feel X". For me, this was
"stuff that makes me feel creative/unique". I found that anytime I was doing "stuff that makes me feel creative/unique" I became "completely engrossed in my work and won't sleep, eat, etc. until it's finished".

SUMMARY - Some tasks/activities/stuff can organically motivate you because it makes you feel a certain way

PART 2 of 3

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Now, there are some tasks that were towards the middle of the spectrum ("stuff that kind of relates to X or gives me some feeling of X every once in a while"). But then, and this is the dangerous part, there were tasks that were the opposite of X. So let's say Y = -X and call it "work that makes me feel like Y". Y work was terrible. Y work made me question whether I should be in the industry and even made me feel depressed. So, in order to avoid these bad feelings, I would "procrastinate terribly [which in turn] accomplishes nothing but adding more stress to my life".

This is where you start to go nuts. You feel like its lose/lose (hate my life when doing it / stressed out when I don't). Sounds like this is the point you may be at. But as I eluded to before, the avoidance of these feelings makes sense. Logically, if something feels bad, we will either put an end to it or try to not let it happen in the first place. But let's dive into these feelings a little bit more. If you are not sure what kind of feelings this Y work can cause. Just ask yourself this question:

"What would happen to me if I had a job where I did nothing but Y work"?

Obviously, its much more effective if you fill in the Y. Again, as an example, for me the question was: "what would happen if I had a job where I did nothing but fix other people's bugs, uncreative maintenance work and writing a bunch of documentation?" My answer was (and this is where is gets very 'psychological'): "I would feel like I am not contributing anything special to the organization. Then I would feel I wasn't unique. Then I might be lost in the crowd, forgotten or abandoned". Everyone answers will be different, but in most cases it fits a model of "I would initially feel _________, but that might lead to ___________, which just feels REALLY scary for some reason". The second 'blank' is usually one of the basic fears: unworthiness, abandonment, loss of safety, ill health, etc. I found that being able to honestly and accurately complete that sentence exposed many subtle, but clearly driving forces in my everyday and long term decision making.

SUMMARY - Answer the question: "What would happen to me if I had a job where I did nothing but Y work"?

PART 3 of 3

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Once you see that some work "makes you feel Y", and that Y feelings suck, its easy to understand why procrastination might happen. But because some of these tasks seem necessary (I'm afraid it's going to cause me to fail school), you try to find a way to do them. So let's address your specific question: "How do I make otherwise un-motivating work intrinsically motivating".

I would say you could accomplish this to two degrees:

* first degree - somehow believe that doing Y work will NOT result in something that seems terrible (answer to SUMMARY question above)

* second degree - figure out how Y work can let you feel traces or bits of X

Imagine, abstractly, that the first or second degree is possible. Do you think you would procrastinate less? I imagine you would because you would not be avoiding what seems like the realization of something terrible.

Now, these two suggestions don't seem very actionable (i.e., 'okay now what?'). Things you can actually do or take action on usually results in a change of behaviors and patterns. Unfortunately, behaviors and patterns become so specific to a given person that to give a general diagnoses would be a waste of your time. Thus, you have to find the change that works for you. I will continue to provide a personal example, but be sure to read the summary at the end as it will give you a more specific question to be asking yourself and others.

My major catalyst turned out to be a "positive vs. negative framing" situation. I framed all of my Y work as "negative." Anytime I would be assigned something in the Y work category I would have quick and subtle thoughts run through my mind, "Great... now I have to put up with this crap again. This work is so meaningless and doesn't even make sense. What I really want to work on is X. But this Y work is holding me back... how dare you Y work!".

Framing my current or future situation in this negative way didn't allow me to provide myself with any "intrinsic motivation". Instead my motivation was thinking about all of the cool X work I could be doing. By simply identifying the framing of my thinking, catching myself in the "quick and subtle" negative thought process and a bit of self-coaching, my stress level (and thus happiness) shifted dramatically! Doing these tasks became much easier. The reason this happened was because I was able to stop my "quick and subtle negative thoughts" and tell myself things more along the lines of "okay, so it looks like we were assigned some Y work. Is Y work really that bad? It is going to kill me. Probably not. In fact, I can't believe they are paying me to do Y work. Oh look it's sunny outside!". Again, this was a behavior or pattern that was specific to me. Yours may not be the same, but your outcome should be.

SUMMARY - find out what behaviors or patterns are holding you back from achieving the "first or second degree" and change them

TIME TO START

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So, in summary, look into what about the Y work really makes you avoid it. Then, determine if what you are avoiding is as bad as it seems. It is at this point that honesty really pays off (Personally, I could have easily answered this question with more "quick and subtle negative thoughts", but that would defeat the purpose). Finally if it doesn't seem worth avoiding, change your pattern or your behavior to something that serves you better. To provide some encouragement, I have been able to achieve the "second degree" with my job. I literally went from despising it to enjoy it! It seems impossible, but it's true. I could really have just responded with the comment "Oh hey, that sucks. You should do more positive framing". But my guess is that would not have helped you that much. Instead, the change needs to be specific to you.

It's easy to just shove these kind of things to the side, but situations like this can cause a great deal of stress and unhappiness. I am sorry that you are currently experiencing it. But, my advice would be instead of taking a large risk: switching jobs, switching careers, dropping out of school, etc., try taking a small risk: asking yourself if what you are avoiding is worth the stress of avoidance.