If your wood is very thick, and the posts will not reach, there are two solutions. The traditional solution is to counterbore in order to reach the post. This drawing shows a counterbored post. Period posts or cotter pins were often too short; and early cabinetmakers were forced to chisel in order to reduce the thickness of the wood where the post passes through. Today's cabinetmaker might prefer a spade or Forstner bit.
The second solution is to use sex bolt. These are internally threaded bolts which spin on the post from the inside of the drawer. You can order sex bolts from Bolt Depot