Not to nitpick, but driving off DMS is only one purpose of the boil, not the whole purpose. There's also killing bacteria, hop utilization, and allowing Maillard reactions to consider. Probably a few other things too.
To the OP, no you can't get too much aeration from shaking, or even from an air stone if you're just using air. I'm not 100% sure on the numbers, but I think it's something like a maximum of 5ppm oxygen with shaking, a maximum of 8ppm using an air stone, and 8-10ppm is ideal for yeast propagation. Those upper ranges can only be reached or exceeded using oxygen rather than air.
Can you describe the off-flavor you're getting? It could be as simple as not letting the beer age long enough before drinking, or having too high of a fermentation temperature, or the recipe.