Solder paste stencil application
First, for these small apertures, a 3-mil stencil thickness would work better than the 5-mil. OSH Stencils offers both. The extra thickness adheres more to the solder, lifting it from the PCB. Also, even when it works, the thicker stencil can leave too much paste on the pads.

Other than that, I recommend a few things:


Surround your pcb with other material of the same height. Other blank PCBs are good for this. This keeps the flexible stencil from bending down around the edges, lifting away from the surface of the board.
Tape these supports to the table with masking tape or Kapton. Tape the PCB to the supports. Tape the stencil to the supports. Tape, tape, tape!
Make a thick line of solder along the left side of the board. This assumes you are right-handed and will drag the squeegee from left to right. Don't bother putting paste down near each hole.
Drag the paste across the board in one motion. As you go, make the angle of the squeegee more and more acute. Start around 45 degrees and end up at 25-30 degrees. This forces more solder towards the holes.
Peel off the stencil.


If you end up with a few holes unfilled, I would remove the stencil anyway and apply paste manually with the syringe. Swiping over the stencil again often messes up the previously-good solder deposits.

Also, if your board is small, you might find that a utility knife blade makes a better squeegee than the supplied plastic card.