Thanks for your very good description of your pain.
It is impossible to tell between the slipped rib or muscle/cartilage injury without a chest x-ray. It is easier to tell if several ribs are fractured by the movement of the chest wall but that does not appear to be your case at all! Thank goodness!
I would continue RICE therapy but also see a chiropractor for treatment. since it has been this long and you are experiencing acute pain like that.
Also you need to give those muscles rest for at least 2 weeks then gentle strain for another 2 weeks until more healing can take place.
You are correct about the length of time it takes to heal being several weeks.
Even if it was a rib fracture then there would not be a lot that your doc will do other than RICE and limitations as described above.