Carpentry and Woodworking - Oak Plug too small - any advice

I am installing new handrails and newels on my basement stairs. I drilled a 1" hole in my handrail and newel for installation and now the 1" oak plug I bought does not fit. It's too small. I must not have kept the drill bit steady enough with my portal drill (I don't have a drill press). Any suggestions on how to fill the hole? Can I make or buy a plug that is a bit bigger than 1"? The same thing happened with a few 3/8" holes but I am ok with using a "mushroom" style plug to fill these. The 1" newel holes are exposed and I want to sand flush with newel.

Thanks!!!

Try mixing a bunch of oak sawdust with a little carpenter's glue. Roll the plug around in the mixture to get a good coating and push it into place. Once it's dried you can sand it flush.

You could mix up some water putty and ad stain to it. Home Depot carries DAP brand as well as Durham, they both work but you can satin the DAP after it dries which might be better.

In your post you mentioned making a plug. If you have some basic woodworking skills it shouldn't be hard to do. Use a compass to draw a circle slightly larger than the hole to be filled. Cut it out with a bandsaw if you have one, or a jig saw. Use a sharp chisel or knife to pare the plug down. The easiest way is to make it slightly tapered like a cork and trial fit as you go. When you get a good fit glue it in, then sand flush and finish. (Oops, I just noticed this post is almost 3 weeks old, so it's probably too late for my suggestion).

Not sure if they make 1-1/8" plug cutters but that is the first thought I had. My second was to use a 1-1/4" hole saw then enlarge the quarter inch pilot hole in the cutout just enough to take a 3/8" plug.