If you don’t want to cut your hair, you may need to accept that it will take several days to clean it properly. It will take patience. Don’t rush. If you have to go out, wrap your hair in a turban or coil it on top of your head and put a beanie on. I would suggest sleeping with a beanie on, too.
Rather than soak your whole head in any treatment, work on small amounts of hair at a time. That way you will also be able to trial treatments and see whether it’s worth continuing.
Oil is probably the best first thing to try. But oil generally works better on dry things, so don’t be too disappointed if it does not clear your hair immediately. Try it on a damp section tonight, and on a dry section tomorrow. Try olive oil and try baby oil.
Other things to try are conditioner and E45. Work a generous amount of conditioner, and a smaller amount of E45, into separate small sections.
Now try combing these sections through with first a wide comb, then a finer one.
Test everything from the ends up. Leave mid-length and then roots til last.
The trouble is that most solvents are very harsh on hair and could damage it. Oil and E45 are a pain to wash out, but will not harm your hair.
You have my sympathies. Dd used to be able to sit on her hair. One lovely summer day she went swimming in a pond, ‘mermaiding’ with her hair loose. Her hair tangled up hideously. Worse, it was full of moss and pondweed, which dried like velcro. In the end she put on her wetsuit and sat in the bath while I treated her hair one 6” section at a time: first rubbing conditioner into short lengths, then combing with afro comb, wide comb, regular comb, finally nit comb. And then rinsing each little section. 6” sections when her hair was nearly 3ft long! It took us all day - hence the wetsuit to keep dd warm. At bedtime I didn’t worry about rinsing her hair, just blotted moisture out and coiled her hair. She slept in a beanie and we completed the job the following day.