Yeah, reading through your updated description of the pain, it does sound like a disc and/or nerve is involved. (I have a bulging disc in my neck from whiplash injury.) The phrase "down the sides of my hips to my joints" is what sounds like nerve-related pain to me. In other words, it is travelling along a major nerve.
My neurologist I see for my neck said that shooting pain, tingling or pins and needles are all indicative that a nerve is involved somehow. In my case, the bulging disc puts pressure on the root nerve, and I get tingling feelings down my left arm. When I had sciatica when pg, the deep muscle progresterone shots I had to take started irritating the sciatic nerve, and I definitely had a shooting pain with that. I had no back pain at all. (Makes sense, since the progesterone shots were done in the hip, not the back.)
I'd forgotten about my chest pain. I also had chest pain about a year ago (pre-dx), again related to the accident. I would sit down and feel like someone had put my chest in a vise. The pain would originate in my neck, then wrap around my chest, like a big squeeze, hard enough to take my breath away. It sometimes lated 10-30 sec. Turns out that was muscle spasms - needed deep stretches in physical therapy to get rid of it. But I also needed to see a neuro to get it dx'd properly. I had been seeing an orthopedic, but he ended up referring me to the neuro on that one.
Rather than endure, I would see someone. There may be something simple you can do for relief, like lay a certain way in bed to take pressure off the area (worked for sciatica). Or, take muscle relaxers to stop the spasms. Or get cortisone shots. Anyway, don't suffer because you think it's a TKI side effect that you can't change. You might have simply injured something lugging the kids around.