You have two problems, disease and insect. The mummified fruit last year was affected by brown rot disease. You need to pick up all those mummied fruit and throw them out with trash to lessen the severity of fungus that cause brown rot. You need to spray you tree with Monterey Fungi Fighter. Follow the label on when to spray the fruit.

The clear ooze you see now causes by insects. The commom insects that bore into plums, peaches and apricots are Oriental Fruit Moths, plum curculio and coddling moths. You need to pick of those fruit. Once those insects lay eggs on the fruit. Eggs will hatch and larvae bore into the fruit and eat seeds or area around the seeds. If you cut open those fruit, you can see tunnels or even worms inside. Any fruit that have worms will drop before they ripen. You need to spray with insecticide.

You are encountering common diseases and pests known to stone fruit (peaches, plums, apricots, cherries). If you do not spray, it will be very difficult to get any good fruit.

Google university extension services like Cornell, Pen State have info for home growers. Look for info about plum pests and diseases. You will find a lot of useful info.

Thank you very much for the detailed answers! Will do and I hope I can save at least some of it.