Well after 3 days with the nipple shield, things have improved! The first time I used it, he ate for 17 minutes on one side with no break and then another 6 on the other. The LC says that the nipple shield will give a constant flow instead of a fast flow initially and then a slower flow after letdown. And that's totally what's happening because his sucking and swallowing rhythm is consistent. Not every feed goes so smoothly, but way better than before. Sometimes when he does unlatch a few times, I will burp or take a short break (like 3 min) and go back and usually during that time he is anxious to get back to eating and does much better.
A new problem popped up though- he is biting the nipple shield. When he starts to do this, I take it off and he continues eating fine without it. This is usually after the initial letdown so the flow is pretty much the same without the shield. I was hesitant to get the nipple shield because I was worried it wouldn't be "breastfeeding" without the skin to skin contact, but it's not that bad and like I said, I sometimes finish a feed without it so we still get that bonding!
I would recommend giving it a try at least to see if things improve!