I'm texturized and I want to minimize damage too!
What I do is do a wet set on damp hair and air dry, which straightens the hair out a bit (for naturals too!) Drying your hair in a wet wrap helps too, but I don't like wrapping wet hair because I have to brush it... maybe the ponytail method would work.
Then press (or flatiron/blowdry) the hair with less heat because it's already partially straight.
Also, I don't know if this is because I'm texturized or not, but if I do a tight set and get under a good medium-hot hooded dryer,my hair comes out straight.
That's the least amount of damage I can think of doing...