I do not own a serger. When I made these bags, I finished the inside edges with a "fake serge" stitch on my Janome 6125 QC sewing machine. I've been asked by another Janome user exactly what this stitch is and how I do it. The specifics below will apply only to my machine, but I'll try to be general too so you can apply it to your machine.
I found out about the "fake serge" in my sewing machine's manual under the utility stitches section. It wasn't called that and didn't say that the stitch could be used in place of a serger, but the description sounded promising as it was to "finish a raw edge of the fabric to prevent fraying." My machine came with an overcasting foot (foot c), which looks a lot like a 1/4 inch foot with the metal guide at one end but with wires running through the middle. There are two stitches that use this foot, overcast (which is just the zig zag stitch #3 using the overcasting foot) and overedge (stitch #19 which you only use with the overcasting foot). Both allow you to finish raw edges, but the overedge also seams so it probably acts more like a serger. I prefer the overcast because you can adjust the width and length slightly, though I prefer the shorter length to the longer one, as you end up with stitches closer together and less fabric showing. The overedge stitch, which looks a lot like a blanket stitch, has a fixed width and length. When you use the overcast foot, be sure the stitch width is wider than the wires, otherwise your needle will hit those wires and either bend, break or damage the foot. I've found that both of these stitches work and look better on two layers of fabric than a single layer, so sew your seam first then overcast or overedge.
It is good to read (or reread) you manual from time to time, and to experiment with feet and stitches on scraps of fabric. Even though I use the same stitch and foot 98% of the time, sometimes those fancy stitches and weird looking feet are be useful. And label those fabric scraps! I have a whole pile next to my machine with the stitch number, foot name, stitch width, and length. Whenever I need to decide how a zig zag stitch should look, I pull out the scraps, find the perfect one, and plug the numbers into my machine.