How do I hang pictures up on textured wall without using nails?

You can try command strips but I would not trust it.

command hooks may work for you

Maybe museum putty

Not much will stick to textured walls unless it is not textured much. I have minimally textured walls, probably from painting for 64 years on the walls, but I used command strips to hold up our sound system speakers. They are small, but weigh a lot for their size. I purchased the weight that was over the weight of the speakers and used two or more depending on the size of what I was hanging. I have never had one fall off the wall yet, the only time I had to put new ones on was when I had to move two of the speakers due to a larger TV. The other three have stayed in place for going on five years now. I think the strips I got were for about twice the weight of the speakers, I wanted to make sure the strips could handle the weight of the little speakers for the small size. You could do a trial and error and see if it would work if you have a picture you wanted to hang that is a heavier one without glass, that way if it falls, glass won't break. Make sure to follow the directions on wait times before hanging the pictures. Holes in the walls from nails are easy to repair and usually unnoticeable when filled because they are so small, even on textured walls.

How about one of those wire art hanging systems?

Hand sewing needles will hold up to 10 lbs, they do the least damage. That is what I have used. Even sewing pins will hold light weight pictures and frames