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Yes, they do continue to ripen off of the tree.

If you want to pick them a little early, it will not hurt anything. Of course, the best is when it is picked at peak ripeness. When this is not possible due to pests, such as bats, then picking them early will give you, at least, some fruit to eat.

You can try bagging your fruit, which may help to deter the bats. This article will help: https://www.gardeningknowhow.com/edible/fruits/fegen/bagging-fruit-trees.htm

They bloom early, perhaps you missed it? And they bloom before new leaves emerge. They often are nipped by late frosts.

https://www.gardeningknowhow.com/edible/fruits/apricots/apricot-tree-growing.htm

They do appear to be dwarfs. It isn't uncommon for them to reach those heights.

As far as what is going on with the fruit- This appears to be Brown Rot disease. This will be a lifelong disease that will not leave the trees. Without yearly treatment, and careful pruning on a strict schedule, it will never produce another edible crop.

You may try to contact your local extension service. They can offer more advice, and help you to craft a plan to maintain the trees.

This link will help you to find the closest extension service to you: https://www.gardeningknowhow.com/extension-search

These articles will offer more information on Brown Rot:

https://www.gardeningknowhow.com/edible/fruits/apricots/diseases-of-apricots.htm

https://www.gardeningknowhow.com/edible/fruits/peach/brown-rot-fungus.htm

First determine if the gummosis is caused by disease or not. Scrape away some of the gum to see the color of the inner bark. If it is dark, the gummosis is caused by a pathogen. If it is healthy, cream colored, there is nothing you need to do. If it is disease, these articles should help. Cultural controls may be your only recourse. I also would contact the local cooperative extension agent for advice. If the disease is severe, they will probably suggest removal of the tree to prevent spreading the spores.

https://pestadvisories.usu.edu/2017/04/18/gummosis/

https://www.gardeningknowhow.com/edible/fruits/apricots/apricot-with-leucostoma-canker.htm

Apricots can be a little finicky. This is especially true in your zone, and warmer. They require around 700 chill hours to bloom.

There are a few other things that could be wrong, too. This article will help: https://www.gardeningknowhow.com/edible/fruits/apricots/apricot-not-blooming.htm

It is part of the infection that is causing the gummosis. Gummosis is a broad term that describes any number if afflictions that can cause sap leaking.

Depending on the infection, it may not be able to be cured, but only managed if done perfectly.

If you have treated with the proper fungicide for the exact disease with no luck, then it may not be curable. Some tree diseases are permanent similar to some human diseases.

I would contact your local extension service if you have not tested to see which disease that your tree has, exactly. In some cases, without knowing this information, it will be near impossible to cure the tree.

This page will help you to find the closest to you:

https://www.gardeningknowhow.com/extension-search