Any apple tree specialists?

When I bought the house, it came with two apple trees. I’ve looked at the Ag Depart. guidlines, but they haven’t helped me much. The trees have fruited every year but never manage to keep the apples on the tree long enough to ripen. The fruit either drops off when small or turns brown and mushy on the branch. And this year, I have entire branches dying and turning brown. I don’t see any visible parasites, although there are some, um, sappy/sticky areas on the dead branches. I don’t like to use chemicals in my yard, but I’m worried about the trees. Any ideas?