You don't have to cut the yarn each time. You can run all your different color yarns up the right side of your work as you progress. You can twist the opposite color around the yarn color of the first stitch to weave this in as you go. You can also just run them up the side and leave the strands long.
I sometimes crochet a border around knitted blankets, and then I'll hook these threads into the border as I go. If you're not into crochet, you can knit a border around the fair isle blanket when you're done. Pick up stitches on each of the two sides and knit, then pick up and knit stitches on the top and bottom edges. A simple garter or seed stitch border will help the stockinette edges from curling. Again, you'll be able to incorporate those threads on the right side into the border as you pick up stitches.