For the DNA I made, I imported pdb data and generated curves from there, but I'm not suggesting you do that. Probably overkill for a logo.
But for accurate DNA, I do recommend you make sure the helix is right-handed (Make a thumbs up sign with your right hand and follow the helix in the direction of your thumb; your fingers should curve in the direction of the spiral. The helix you have made in post 5 is indeed right-handed) Also beneficial to incorporate a major and minor groove, instead of perfectly offset helices.
It doesn't have to be extremely accurate, it's just for the circles that will spin around the atom like wheels but also around the atom's axis.