An LC will definitely be able to help you. Do you have a pump? I'm betting the engorgement is probably the main culprit - but you should also check the baby for tongue-tie (wait til he screams and look at his tongue to see if it's flat or heart-shaped across the tip). Cup-feeding anything you express will help too.
As for the engorgement and possible mastitis - take a hot shower and get a fairly wide-toothed comb and a lot of soap. In the shower, gently but firmly "comb" your breasts (using the soap to lubricate) from the far outside parts (well under your arms) towards your nipples. Express as much as you can every few strokes. That should help loosen everything up and prevent more clogging. Then try to nurse ASAP after the shower. I don't see it used a lot here, but you could try an "australian" style latch - wait for an open mouth, then manually place the lower line of your aereola on the baby's lower lip and fold/stuff/unroll the top of the nipple as far back into the baby's mouth as you can. It's a little more interventiony than is typically practiced but I assume you're well-versed in the normal baby-led latching and if that's not working, you might try something different. Another option might be a nipple shield? They're not ideal but they're better than bottles, bleeding nipples and loads of pain.