Burfab or Veltrex is a great fabric to use when you want something to have a smooth almost suede-like finish. I also think it looks a lot like flocking (but much less dangerous to work with).
The only down side of Burfab is that it's next to impossible to hide your seams. I recommend trying to place your seams in spots that will later be hidden by something else going on your mascot/puppet (e.g. under the arms of glasses or moustache), or in a more natural spot like a laugh line or forehead brow line.
Once you have the Burfab pattern figured out, machine sewn together, and glued over your shape it might look rather...well...sloppy! Don't worry! Here's my favorite part! STEAMING!
As you can see in the image below, this mascot shoe looked pretty rough (on the left) when it was first covered in black Burfab. But after the magic of a bit of steam it looks smooth and hugs the shape beautifully!
Once the Burfab is steamed smooth, all you have to do is go over it with a sticky lint roller and it will look incredible!
Now get building!