Use a 3/4" nap lambswool roller. Your crappy little roller is pulling the paint off the wall. It may seem counter intuitive, but that's the key to a good professional job. Take the roller out of the bag, wet it down in the sink, then wring in out. Put it on the frame, roll it slowly into the paint pan, allowing the paint to absorb into the roller. Back it out and get rid of a little bit of excess, Then using a roller with and extension, start at the bottom of the wall and go up as high as you can and then back down. Don't go back and forth over the area you already painted, maybe go over it once to get excess paint off.
This type of roller will hold lots of paint and lay it down in a thick enough layer that you won't be doing more than 2 coats. Most problems with painting are the operator and tools, not the paint.
I have had lambswool rollers last for years. Just clean them out well and store them. Be patient when cleaning. They are not throw away items. Don't let them sit around while you take a break either. Wrap it with a plastic bag so the paint doesn't dry out especially with acrylics. They dry out quickly.