Jim, do you have a small, sharp hand saw?
Do you understand about branch collar cuts? 3 point pruning cuts? Thinning versus heading cuts? A little time on google will inform a lot.
Sometimes, branches that cross are supporting each other, if they have been growing that way for a long time. Remove a supporting limb, and the other may fail.
Little bits at a time, and step back and walk around the tree to look. Stroke you beard as you look at it. It will look like the master mulling over the situation. The tree will be convinced. They're easy to trick. The un-stressed apple tree grows better, just like a cow.
Okay, not really.