Welcome to Knitting Help!
I'm probably not understanding the problem. When you change the size needle, the larger needle is in your right hand and the smaller needle is in your left. The working yarn is coming off the smaller right needle but you just pick up the yarn and continue to knit stitches from left needle to right. When you finish the round the working yarn should still be coming off the right needle and you can pick up the other end of the larger circular in your left hand and continue knitting. Can you do this? Pick up the other end of the larger needle and continue knitting? or have things somehow gotten reversed?