Picking up stitches to knit a border works well, but it won't entirely hide bad looking edges, they will still be visible in the "seam" made by the pick up, but will be on the back side at least.
I wonder if a person could do kind of an edge like might be put on a quilt, but knit style. (I've never heard of it or tried it, just an idea) I'm thinking pick up the stitches knit a number of rows and make a turning row maybe (with a purl row on the outside, or a eyelet row yo, k2tog around), knit a few more rows then fold it clear over the problem edge and whip stitch it down on the other side. I don't know what would happen at the corners. Possibly do each side separately?
A crochet border might help hide them as well. Just brainstorming here.
Whatever you do if it involves picking up stitches you usually need to skip some of the rows up the side when you pick up, and do the stitches on the ends stitch for stitch to get it to lay smooth. I'm not sure the the ration of stitches to pick up is in bulky yarn. Check into that before you embark.