I sure don't understand why it would hurt a machine to lay on its side. There is nothing to leak.
I took an IKEA SY sewing machine to Lithuania last summer. Weighs about 13 pounds. I put a few clothes in the bottom of the suitcase, laid the machine on top of that; stuffed the throat cavity with more clothing; and then took the straps that are inside most suitcases and securely tightened it down so it would not move around. Then I added other things that would fit. It got there perfectly. I just couldn't add too much other heavy stuff because of the weight limits. Clothes were the best bet. If you are taking heavy shoes or other clothes, WEAR IT.
You could also put a machine in your "purse". There are some really nice wheeled bags that will hold a machine, and you can stuff it around the machine with "essentials" and call it your purse. It would need to fit under the seat. We flew United last week and there was a lot of room under those seats. Very comfortable.