Yes, I can see where cutting/tying each time would be a total pain.
Why don't you just, as you suggest, carry the colors along the edges - then put an edging along the scarf after you've finished it? Then you could hide the carried yarn strands.
This edging could be either knitted - pick up stitches and knit them, then quickly bind off - or crocheted, if you have that skill under your belt. Or maybe you could just sew a whip-stitch along each edge.
My words sell the Magic of a Dream