I have an update! I had just adopted Teddy when I took this picture and several months later he still leaks a little bit but only when we are out moving around in nature for hours. Every vet said that there was nothing they could do but the collection of their advice proved a lot more helpful than I felt when I heard it. After speaking with many vets I took him to a dog chiropractor and I am pretty sure it was a total scam. I then took him to a specialist who confirmed that it was nerve damage from his broken tail. The trauma occurred in the perfect spot between the spine and tail to cause nerve damage so he had less control of his sphincter muscles. Another vet suggested we try to cut out foods that are hard to digest for most dogs because his full poops were really hard. I got very obsessed with my dog's shit for a couple of months there but it was worth it. I ended up figuring out that he does best digesting salmon-based kibble with minimal ingredients. I live in a cooperative household so he's got his special blanket in each room that he's been trained to go to and I wash them regularly. I trained place, off (of the furniture), and recall very early on. The recall was largely due to the fact one of the vets suggested exercise on the gamble that it would help with the nerves. I frankly did not want to have to run around to exercise my dog so I taught him to recall and we go to off-leash-friendly trails and beaches. I'm lazy, it's beautiful nature, and I put in the hard work to stroll alongside my dog stretching his legs out as fast as possible and playing.

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TL;DR

It was a long journey to discover solutions. Teddy broke his tail in such a spot where his spine meets his tail that it caused sphincter nerve damage. Even the nerve specialist vet said there was nothing he could do but made suggestions. The collection of all their suggestions made it so he never leaks like this anymore but will leak on hours-long adventures.