Clay takes longer to drain right through (percolate) compared with sandy soil.
You will need to regrade the back yard so it is a tad higher at the house and a tad lower further back.
Why the front yard has problems while it is already sloped awy from the house I don't know, unless it has undulations as opposd to being a smooth slope.
The disadvantages of crab apple trees. In summer, the apples are too sour to pick and eat. In winter the birds come and leave dropping all over the place.