I always play every 3DS game with the Full 3D on. Never once had I ever had an issue with it. And used to play my 3DS a ton. I still play it a lot, however, ever since the Wii U game out, that's gotten me back into console gaming. So I've put a lot of time into that. But before that, I put tons of time into my 3DS. And again, I always play with the 3D on full. No problem at all. Sorry that you are having such issues, like dizziness, and perhaps anything else. But I've never had any issues or problems come from constantly playing in full 3D.
Of course, some games "3D" is more apparent than others. Some games tack it on as an after thought, so the 3D effect isn't that great. Other games build the game around the 3D effect, and so those games look fantastic with full 3D. So it depends on the game.
Then, it also depends on the kinds of 3D as well. I find that I like the 3D where it pulls back away from you, to give you depth, far more than I like the 3D that pops out at you. Super Mario 3D Land actually gave you the option of both. It allowed you to choose whether to 'pop' the 3D at you, or 'pull' the 3D away from you. I like the pulling back way better, as it gives more depth, and looks nicer. I think. The popping out at you can be nice, but other times, it doesn't look as good as if they would have just pulled it away. So, maybe that's your issue. Is your game popping the 3D at you, or pulling it back from you. Also, a friend of mine, who wears glasses, has issues with the 3D, but it's all 3D, even movies, and not just the 3DS. It's just his eyes. However, he found that if he's starting to have an issue with the 3DS, all he needs to do is just blink a lot, really fast, and he'll get re-adjusted to the game and then everything will be fine again. So you can try that trick.
Other than that, I don't know what to tell you, other than sorry that you are having those issues. Hopefully it passes. If not, an eye doctor may be in order. That, or you may just have to enjoy the games without 3D. Either way, good luck.