I had this problem whilst breastfeeding my 2nd DS (2 years ago now).
We stopped BF at 7 1/2 months and even though he was only on 4 feeds (ish) a day, I still filled up so I looked like Dolly Parton! I had to wait it out and, YES, it is EXTREMELY painful.
You can take paracetamol and ibuprofen - 4 hours apart but don't take them together, there needs to be a 2 hour separation. So 8am paracetamol and 10am ibuprofen then 12noon paracetamol and 2pm ibuprofen etc, make sure you don't exceed the dosage and I used to try and space them out so I was taking them in the middle of the night as well.
I also tried EPSOM SALTS which a lot of people told me about, I took three lots per day. Is supposed to dry your milk up quicker but I don't really know how well it really worked.
Finally I put some savoy cabbage leaves in my bra to help relieve the pain but again, I don't really remember if this helped much.
It's quite hard to tell what worked for me realistically - all I remember is that it took 2 weeks for my milk to start drying up and everyone said it would be over in a week for me and it wasn't
I do remember being told NOT to express (and I didn't) but I did put those avent breast shells into my bra to help relieve the engorgement, maybe for 10 mins a day until it was enough to bear.
HTH - Kez