This probably depends in part on the size (eitz to eitz length, I mean) and weight of the sefer torah, but here's what people in my minyan do (and what I do personally) with a medium-sized scroll. My description is for a left-handed person, but you can reverse it.
Sit with your back against the back of the chair. Place your left hand in front of you and under the "disks" at the bottom end of the eitz, so you're supporting weight without being poked. Let the weight of the sefer lean against your left shoulder. The sefer is not vertical; it's leaning against your shoulder, bottom out a bit. (It's also slightly slanted across your body.) For our scroll this does not result in an eitz poking you in the thigh; the distance between you hand under the disk and the end of the eitz is short enough that the eitzim themselves don't rest on anything. If you need more stability, place your right hand on the front near the center.
Sitting like this, it's easy to shift from left to right -- start by replacing the bottom hand, then slide your now-free left hand up to the middle of the scroll and shift the scroll from your left shoulder to your right. (This does assume that you don't have a book in your free hand at the time.)
If you have any special medical conditions that affect your ability to bear weight in certain places, you should of course consult your local physician.