it's entirely up to you really, you can carry them up the sides, no worries, but i only do this when i am doing an item that will be seamed, to hide the stranding - i know it's not hugely obvious, but i know it's there!!. Since it's a scarf, personally, and i know it's a lot of weaving (i've been taught to have a more positive attitude towards finishing than a lot of other people round here have
), but i would do the weaving in.
Prefer to spend that time, than have to look at those nasty strands for the rest of my life!! So it's up to you.