Once a dowel snaps off in a joint or breaks along its length, there is no easy way to pull it out. Carpenter's glue actually dries to form a joint that is stronger than the wood itself. While many glues are reversible, provided you know what kind of glue was used, getting into a tight spot to dissolve the glue and get the dowel out may be a lot more work than it is worth.
Cut the dowel off flush with the surface of the object it is glued into, using the saw, for ease of drilling.
Draw an "X" across the widest diameters of the dowel to find the centre point.
Drill a small pilot hole using a small bit (around 1/8-inch) into the centre of the "X" for a precision job.
Drill down into the dowel using a bit the same width of the dowel. The pilot hole should get you started on the right course. Be careful not to drill too far; use the variable speed if the drill has it, and check you progress every few seconds as you get deeper into the dowel.
Clean out hole, depending on size, with a bit of rag or canned air.