Hi, Jodie,
I have always sewn the bodice and bodice lining to the skirt as one piece and then serged the seam to finish. Since you don’t have a serger or a zig-zag stitch, you could finish the seam with bias binding (either pre-packaged or made from the same fabric as the flat piping). Or you could attach just the outer bodice fabric to the skirt and then finish the raw edge of the lining by folding it under 1/2 inch and hand-stitching it in place to cover the seam allowances of the bodice/skirt seam. Or (if you hate hand-sewing as much as I do) you could fold under the raw edge of the lining, hold it in place with pins or double-sided water-soluble tape, and then top-stitch from the front, catching your folded edge for a neat finish.
This is off the top of my head, so others may have better ideas or at least more coherent ones!
Edited to Add: I bet your dress looks lovely with the french seams. I love when the inside of a garment is as pretty as the outside.