I have a walking foot for my Singer 201, it's a straight stitch only model and doesn't have the overlock and over cast patterns you might want on jersey. I use it mostly when sewing layers of fabric, if they aren't pinned well enough or very slippery they might sort of go out of alignment. The attachment goes on when needed, and comes off for regular jobs.
I have an old Singer 99 I can sew light weight jersey, even flimsy viscose-cotton-elastane blends. It's the most basic machine possible, straight stitch only, but when I lower the presser bar tension it stitches most fabrics with out any fuzz. It doesn't stretch the fabric or act up when other macines have. I think it might have something to do with the narrow foot too, I'm not sure. I haven't found a very good explanation for it since most other machines are much more advanced, with all kinds of functions and features in comparison. My point is really, if you play around with settings, perhaps baste by hand, or use more pins there might be away round it. The advantage with an overlocker is mostly the multiple needle stitch patterns with more loops and threads are different to basic machine. An overlocker can act up in much the same ways as the basic sewing machine, especially with slippery strechy fabrics.