Welcome!
As to your question: The first suggestion would be to stabilize the pine cone centers in a vacuum / pressure pot so that the stabilizing gets all the way to the center. You could try cutting the pine cones in 3/4 in squares and then casting them but that would still leave the centers very soft and weak.
The next step involves the drilling side.
1. Make sure you are using good sharp bits.
2. Check your speed. Most people like to use slow speed in drilling but generally, I use faster speeds and slower feed. A fast speed has less tendency to "catch" and more prone to cut through. There is a time and type of wood/material that slower speeds are required.
3. Use a good blank vise. Holding with pliers or hand clamps will allow for movement that will catch and do exactly what you describe. Blank Movement during drilling is the enemy, unless skilled at this process. Also, the more of the blank that is held or gripped in a vise, the less likely it is to blowout. It is easy to drill a 3/8 in hole in a 2X4 but when drilling a 3/8 in hole in a 1 in square block, it is more prone to exploding because there is less structure holding it intact. This is where a good pen vise with a good grip will help, especially from the middle to the exit end.
