Ok, this is a problem lot's of people have, especially if they are tight knitters. So don't feel like you are doing something wrong.
What you can do is use the right needle itself as your push point. Or, get some leather quilting thimbles for the finger you tend to push with. In the meantime (I know how painful those cracks can be
) you might be able to just get away with putting on two bandaids on the affected finger til you can get to the store and get those thimbles. Also another thing is that you could (beings you are throwing n not picking) use the yarn from your right hand and use it to "pull" the stitches off the left needle. That's about all I can think of. I'm sure suzee will be able to help you with different suggestions as well.
Get some good ol' cornhuskers lotion for the cracks in the fingertips. My hubby did concrete and would get those pesky "concrete cracks" in his fingers and that stuff would work wonders for him.