Sure, you can drill, dowel, and drill again.
But for this application, it's overkill in my book.
There's absolutely no tension load on those screws.
I'd dribble some 5-minute epoxy into the holes. Then, you can drive the screws in before it sets if you want for superior hold. You can always apply heat via a soldering iron to the screw heads to soften/de-bond the screws later if you need to. Or, just wait until the epoxy sets up slightly and pre-drill the screws to the dia. of the screw shanks.
And that toothpick trick works great, too...