The epoxy method's not bad but you have the mixing time and hassle.
Here's the cmac method. Buy a supply of dowels from 1/8th up to 3/8th at least, 1/2 inch can be useful sometimes. Cut them into roughly 1 foot lengths -- that way you can haul around a bundle of various sizes rubber-banded together in your tool box. If you don't have a dowel that is a perfectly tight fit in the hole you need to plug, go the next size up, never use one that's slightly loose in the hole.
Get ahold of a roll of
Porter Cable adhesive backed 80 grit sandpaper. Amazon has the price crossed out there but you can get it at most lumber yards for around $15. Worth its weight in Brazilian Rosewood. The stuff is 4 1/2 " wide. I cut blocks -- 1/2 " MDF is the best but plywood will do -- 4 1/2" x 2" approx and wrap three sides. Cut the excess with a utility knife and sand any bit hanging over with an older sanding block. This can also be real handy on one side of a piece of doorskin (1/8 plywood) or 1/4" ply. MDF (medium density fiberboard) is trick, I used to assiduously save scraps of it on jobsites -- 1/4 or 1/2 my favorite. In the long run, you'll spend less on sandpaper with this stuff than sheets cause you're not scrunching it all up while using it. I've tried the lighter grits, 120, 150, and those don't work as well - lose their bite too quick.
Anyway, on the dowell, if it's quite a bit bigger than he hole, I shave it down a bit with my knife and then sand it with the block. Spin it as you sand and put a slight taper on it. When it gets to where you can put it about as far as the depth of the screw, maybe a little less, give it a bit of Titebond and drive it in with a hammer, not too hard. By the time you've cut off the excess with a sharp chisel, the glue will have enough bite to allow you to drill.
I've done it hundreds of times. With that kind of friction/tight fit, Titebond will start to grab right away. For best results, use a 1/16th bit to start the new hole. It's much easier to aim those with accuracy. The bit size you need for your screw, say 1/8, will be much more likely to wander especially with the different characteristics of the original wood vs. the plug/dowell. The 1/8 bit will track the hole the 1/16 bit made.