Here’s one way to do this:
I have also just dropped the old yarn and started knitting with the new strand. When I’m finished I make sure to cross the two ends to avoid a hole and then I split the yarn into plies and weave the plies in separately (a trick from Jan_in_CA).
That is fine but the yarn is not comprised of two strands twisted together so how would one deal with that
What yarn are you using?
To start the new yarn, drop the old and knit with the new strand. When you come to weave in the end, cross the 2 yarn ends to close up any hole. You can check this by crossing the ends and looking on the opposite side to see that there is no hole.
As for weaving in the ends, you can use your favorite method. I like to work this way even with chunky yarn: