It seems that your pieces of wood have shifted somewhat since you started drilling.
If you can eyeball the 3 pieces to see which one(s) seem out of line, use one or more clamps to persuade the wayward one back into position. Once you have lined them back up, you should be able to get the bit out and, without removing the clamps, drive your dowel into the hole.
As an alternative, you could put the drill back on the bit and give it a spin*, and that should cut the sides of the hole to line up with its edges again. Once you've cut the edges of the hole, the bit should come out again. Note that if you do this, you will have at least one elongated hole and when you put the dowel back in, that elongated hole will allow that one piece of wood to remain out of the desired original position.
Whether or not it matters if the 3 pieces are out of alignment when pinned will probably be determined by how much finishing work you're doing after the pinning is done, and how far out of alignment the pieces are.
*If your drill bogs down and won't spin the bit because of the tension of the misaligned wood, you may need to use a long handled wrench to turn the bit, instead. If you have a hex-shafted drill bit, a 1/4" box wrench (spanner) should grab hold of it just fine. Since those tend to be small, a bit of pipe over the other end of the wrench should allow you to generate plenty of torque to turn the bit. If it's a round-shank bit, you may need to use a pipe-wrench or monkey-wrench, however, the serrated teeth, which give you a better bite, may dig into the shank and damage it. Caveat emptor.
