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I would recommend trying neem oil. It is effective against both harmful insects and fungus, and it's very safe to use. Here is more information: https://www.gardeningknowhow.com/problems/pests/neem-oil-uses.htm

Here is an article that will help you that discusses split cherry fruit:

https://www.gardeningknowhow.com/edible/fruits/cherry/fruit-split-in-cherries.htm

Happy gardening!

If it is a well established tree, some temporary root exposure will not hurt the tree much. If it is not too late, you can cover the roots with wet blankets to reduce damage. When they rebury the roots, give the tree some bone meal. This will help it regrow any roots that may have been damaged.

It might be bacterial wetwood, peach borers or canker, depending on how the sap looks. These articles will describe the nature of the running sap for each problem and will suggest solutions:
https://www.gardeningknowhow.com/ornamental/trees/tgen/wetwood-bleeding-trees.htm
https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-borer.htm
https://extension.colostate.edu/pubs/garden/02937.html

My research shows that this is a self-fruitful cultivator and does not need a pollinator.

It might be bacterial wetwood, peach borers or canker, depending on how the sap looks (though I'm leaning on borers as you say it's clear and jelly like). These articles will describe the nature of the running sap for each problem and will suggest solutions:
https://www.gardeningknowhow.com/ornamental/trees/tgen/wetwood-bleeding-trees.htm
https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-borer.htm
https://extension.colostate.edu/pubs/garden/02937.html