I think this is an important question, in particular because there are many other math grad students in the situation of valuing their teaching more than their research and experiencing dissonance because of that. Let's try to take things in some kind of logical sequence.
1) It sounds like your goal is to be a faculty member at a college or university devoted more to teaching than research. If so, you should do what you need to do to achieve that goal.
As I said above, a lot of math grad students are in this position. It does not appear to be that confluent with the structure of a math PhD program, which emphasizes research more than teaching, but times have changed and are continuing to change: the amount that teaching is valued is on the rise. You certainly don't need to hide your intentions; on the contrary, you should express them to your mentors and professional associates.
One of the things you probably need to reach that goal is a PhD in mathematics: in the current job market, the number of "teaching jobs" you can get without a PhD is declining rapidly. So what impedes you from getting your PhD is not a path towards your goal. You need to remember that.
Most importantly, I know that putting this much time, effort, and emotional energy into teaching is taking away from my research. Sometimes I feel terribly guilty after a big exam or project because I realize I haven't even looked at my research in a couple of weeks.
The guilt is only productive insofar as it alerts you to something that you want to change. Indeed it does not seem likely that you can make sufficient progress on your thesis research if you drop it completely for two weeks at a time...but it is possible. More to the point, you need to have a concrete, realistic plan for division of labor between teaching and research. It is totally fine for you to devote significantly more time towards teaching than the average student as long as you still have a workable plan for your research. In my opinion (as a math professor who has supervised several PhD students) a student could get away with devoting, say, 15 hours per week to their thesis research...provided they used that time very productively and intelligently.
Another answer begins
Superficial solution: Control your behavior with a time budget
I wholeheartedly agree. It goes on to suggest 70% research, 30% teaching, which sounds about right on average, but for you might be way too little time on your teaching to make you satisfied. Better to do 30% research, 70% teaching and come up with a corresponding plan. (In case you don't know: there are many, many potential academic workplaces that will not care at all about what you proved in your thesis but will care very much that you wrote one.)
2) An important part of good teaching and good collegial behavior about teaching is flexibility. There are many, many ways to be an effective teacher. You write
I have very strong feelings about the way courses should be taught, graded, and organized.
This is a little worrisome to me: in fact you do not have as much teaching experience as most of your future tenure track colleagues, and it may be too soon to have such strong feelings. Much to learn, you still have.
I have thought extensively about this issue and experiment with my teaching every chance that I get.
And that sounds really great to me. In your interactions with others, I would emphasize your thoughtfulness, your experimentation and your willingness to change practices based on what you learned.
My teaching philosophy is fundamentally at odds with the majority of the lecturers in the department. It makes me feel like a pariah, and privately makes me feel incensed when I see others teaching in a way that I consider unethical.
This really concerns me. I find it unlikely that you are really surrounded by people who are ignorant or uncaring about teaching. I find it extremely unlikely that others in your department are teaching in a way that is unethical: that is a very strong accusation. You seem to yourself feel that you are putting too much time and energy into your teaching, perhaps almost to the point of burning yourself out a bit. That cannot be the ideal state.
3) You portray yourself as an unusually excellent teacher. That's great...if you have corroborating evidence.
Though I'm not vocal about my disagreements, I have been consistently passed over for teaching awards, despite constant effort, innovation, and shining student evaluations. I wish I didn't care about this.
Well, I think you should care. Why haven't you won any teaching awards? This is a tough question, but a key one for someone in your position to ask. I would bring it up with your teaching mentor -- I hope you have a teaching mentor -- or with some other trusted faculty mentor. Are there faculty members in the department who think that you are the best, or one of the best, graduate student
teachers? You really need to find out. (That your evaluations are "shining" is evidence in that direction, but not definitive.)
Let me say that I have known students who throw themselves into their teaching to the point of getting a bit lost in it and neglecting their research. I had one colleague in grad school for whom I always saw students at her office. I asked her what her office hours were, and she told me that they were M-F 9-5. And she was certainly a good teacher rather than a bad one, but so far as I know she was not one of the top teachers in the department, and she may not have won any teaching awards. This brings me to my last point.
4) Just because you are devoting more time to your students does not guarantee you'll be a better teacher in any sense. Part of being an effective university teacher is dealing successfully with students in the relatively limited interaction time you have with them.
Every time I try to pull back and adopt a more traditional model, I get depressed and frustrated, because my students aren't learning anything, I'm as bored as they are, and I wonder what the point of my existence even is if I'm just going to give up and be another trash teacher who goes through the motions all day.
This really concerns me. "Traditional" teaching is certainly effective if done well. When I teach freshman calculus I do so in a quite traditional way, and the time I spend with students is rather limited. Still they are learning something, and in general neither they or I are bored. I know very few teachers who are "trash" or just "go through the motions."
I strongly suspect that there is some middle path between burnout and trash. I suggest that you search for it (with help from others) and try to walk it. Good luck.