Actually the method that Jason describes with a bent nail is effective if there is wet core. I've seen this done with stainless brake cable on an electric drill too. It is flexible enough to bend and get in the core area, yet stiff enough to chip away at the balsa. The key is to get out any spongy balsa and reach dry core. An acetone soaked rag stuffed in there and moved around will help displace any water. If you reach solid balsa but it is wet, then you can use heat and vacuum to dry it out. The syringe and epoxy works for filling. I mix in filler like West Systems 403 Microfibers (white powder) to a mayonnaise consistency. Just remember that when you get too much epoxy and hardener in critical mass, you get a mini "nuclear reactor" that will generate lots of heat as it cures.

If the core is dry when you drill the 1st oversize hole, just do what Jason King describes above.