Best method for painting textured walls?
You just need a little more pressure. It looks like you have enough paint on the roller, and just need to press the nap into the texture more assertively. The texture isn't extraordinarily deep or sharp, and those dots appear to be a matter of nap reach, and not of smearing.

If the latter were the case, you could use a wet re-roll strategy--after you finish a wall, go back and re-roll it right away. I've done this with great success when painting bold colors over light colors, for example.