What kind of acrylic are you using ?
Acrylic Acetate is much more finicky than many other types of acrylic ... heats, melts, drill bit gets stuck, chips, harder, etc. . Rhino plastic is much the same. . The "Pearl" plastics from PSI can be a real PITA.
Alumilite is much easier and smoother and more forgiving ... some say it turns like butter but I don't quite get that analogy !! . Even so, it can overheat, melt and stick to your drill. Alumilite is my "acrylic" of choice.
Drill all acrylics slowly, clearing the swarf frequently.
Use very sharp tools and turn more slowly than with wood; try for those lovely continuous ribbons shooting off. If you can get that, you'll be able to skip the first couple of sanding stages.
Sand lengthwise with the lathe off (turning the headstock by hand) ... 320 up to 2000. Micromesh if you wish (lengthwise motion) but you may not need to. . Polish with Novus 3 & Novus 2, or Huts Plastic Polish, or other plastic polish (Mequiar's automotive polish is very good) ... again the motion should be lengthwise.