H.A.
Hi there, I'm so sorry to hear about your pain. I'm still pregnant with my first ( like to read what might happen after the baby is born and what could come up, so I read a lot of these questions) I read my mom your question and he said she had similar pain and my dad would heat up a wash cloths with hot water and mom would cover her breasts with them for at least ten mimutes before feeding me. She said at first the pain was so intense that she thought she wouldn't get to breastfeed at all. But the washcloths worked!
I've read some of the other posts and agree that it could be a clogged duct, which would explain why hot wash cloths would help, the heat can calm the duct and let milk flow correctly.
I hope this helps, give the wash cloths a try at least, you never know what will do the trick.
Good luck,
H.