Check the pattern directions or charts. You usually use 2 colors per row, or if there are more, they shouldn't need to be carried more than a few stitches. Few stranded patterns will have you work too many colors per row.
It's possible that the pattern is worked intarsia style. You can make some beautiful patterns. Amy has video on changing colors for this technique.
Honestly, though, either way is going to leave a messy back. You'd have to line it, or you could make two squares of each lamb and make a double-thick blanket by sewing them back to back. Or make just colored blocks for the back without the lambs.