This may be stating the obvious, but it wasn't mentioned in the thread.
If you use the sawdust and wood glue mix, two things to consider.
1) use a water proof glue designed for outdoor use. Check the labels but I believe Tightbond Plus yellow is approved for outdoor use. A lot of wood glues are not outdoor approved and when the handle gets wet, the glue will turn to mush.
2) the wine cork idea from Paula is the way I would go. If you don't have wine cork, go to a hobby store (hobby Lobby etc.) and buy a small package of corks. Anytime you're filling a hole in wood (cork is a wood bark) using the same species of wood (maple, walnut, pine, etc.) will help ensure a closer match.
The added benefit of the wine cork is you now must drink the bottle of wine since it no longer has a cork.
The more I type this, the more I like Paula's idea. Skip the Hobby Loboy and go to your local wine shop and get a nice Merlot or Shiraz. Be sure it has real cork for the top (not those plastic faux corks being used on some wines), take a good corkscrew and remove the cork. Cut it deep enough to get past the wine stains, sand some bits of the cork to mix with your glue and patch the holes on your rod.
By now, the wine has had ample time to breathe. What you do next is up to you. Personally, I"d get a glass, fill it up and watch the glue dry...