Fixing crummy kitchen drawers

I am in the process of refurbishing my kitchen cabinets (painting mostly, some repairs, some storage enhancements, some trim). One of the repairs I need to do is to the drawers. They are crummy. Literally. No dovetails, miters here ... badly made, old, crumbling plywood drawers. I'm tired of having sawdust fall on my dishes. See pic and you will know what I mean.

I have a full array of carpentry tools. My Dad wanted to make furniture, so we had all been buying him nice tools for holidays for a few years. Unfortunately, he died (6 years ago) before he even got the chance to use many of them. I know how to use the compound miter saw, scroll saw, jig saw, drill, and hand tools. I am guessing that a router would be the right tool to clean up the edges on these drawers. I have a router, but I've never tried it out. I do have scrap wood to practice on.

I don't know that I have the right bit or not. My plan is to even out the edges and apply some sort of laminate/veneer edging, though I'm not sure what. Better ideas, advice about what type of bit to use, what sort of edge covering to use, good tutorials all gratefully accepted. I'm hoping not to rebuild them entirely, though that is possibly the right answer.

Hello JulieC:
It looks like you have drawers that slide on the cabinet rails with no rolling glides.
To make them work easier and not cause so much sawdust below them, you can rub the bottom edges with a candle. That will lubricate them and stop the sawdust.
From the top, I would use some rock-hard wood filler in the plywood and sand them smooth with a belt sander. If they still look bad to you, you could try painting them, but I don't see any need for the router.
Glenn

You could dress them up as Glenn suggested. They should hold up fine for a few more years with only a couple of bucks spent. Try www.rockler.com for some plastic laminate to stick on the bottom of the drawers, it will help them to slide easier.
You could try a local cabinet shop for some free scraps of laminate...then use some contact cement and glue em up. These areas at the bottom are not seen so it does not matter what the color may be.
Have fun .

Actually, there are rolling metal glides on the sides. Hard to see in the pic, but they are there. Wood filler ... duh ... I'm not sure why I didn't think of that, that stuff is a good friend of mine. :lol

Thanks for that link, Inspector. Plastic Edgemate is about what I was thinking, I just had no idea where to look.