I agree with BeckyW...please go see an IBCLC...not just one of the LCs at a hospital...they may have training but the IBCLCs tend to have more than the 1st line of routine steps to offer you.
I had the issue with forceful letdown/oversupply with my son who is now 11 months. He went on zantac at 3 weeks because of a 3 hour crying spell where I could hear the reflux gurgling at the back of his throat. my daughter was intense, difficult and needy...and i probably had the same breastfeeding issues with her, but I followed some bad advice and reversed the problem with her to a significant undersupply by 8 weeks. She still spit up and was difficult...but more because she was hungry (probably) and colicky. I did give up dairy with her, though.
With my son, I gave up dairy at 2 weeks.He spit up buckets! Did the zantac at 3 weeks. Switched to prevacid around 6 weeks, I think. Went to an IBCLC at 5 weeks because he screamed and fussed a lot, including at feedings. He presented with many of the symptoms that Becky mentioned. My lactation consultant did have my try a position with my son where he was upright and tummy to tummy with me. He would eventually kind of lean to my arm of whichever breast he was on, but honestly, it was amazing when he didn't scream during a feeding that day. As he got bigger things kind of adapted, but then he was also able to handle it better. Another thing that she had me try was this "V" pressure. You take your first 2 fingers and make a V and when you have a letdown you put them on your areola on the top and bottom of your nipple and then press in. Kind of like when you would try to stop your other breast from leaking all over when you have a letdown while nursing. If you're too full, then this will not work so well because it will pull your nipple out of your daughter's mouth. So, if you work on keeping your supply in balance (doing the block feeding) then this works better.
I ended up going off soy around 5 months because he was still having issues with his tummy (after going to the pediatric GI) and that did seem to improve things a little bit more.
I wanted to mention about the elimination diet that it can take more than 2 weeks. From what i was told, it can take 2 weeks to get out of your system and up to 2 more weeks to be out of your baby's system. So, you might want to give it longer. I know dairy and soy are usually the first to try. Then eggs. I'm not sure about wheat...that obviously makes your diet very difficult. Corn is another one. I wouldn't necessarily cut them all out at once. However, you could do that and then add back one at a time. Unfortunately it just takes time, especially if you're dealing with sensitivities and not allergies.
I completely understand the stress, sleep deprivation and the limited diet. Try smoothies with unsweetened almond milk. I don't think almonds are as highly allergenic. There is also oat milk (boxed at health food stores) and coconut milk.
You could try making these bars and control the ingredients that go in: http://www.wholeliving.com/recipe/breakfast-bars-recipe-2.... (You can try oats or barley flakes instead of spelt flakes.) For breakfast, I would make oatmeal with water and and add walnuts and bananas.
You could also snack on hummus (unless your baby is sensitive to the beans) or guacamole and veggies. Also, try quinoa. It's a grain like rice. Cook it in chicken stock for flavor. It is a complete protein (like eggs) so that is helpful too. If you search whole food's site (or in general online) you will find some recipes. I think there are even quinoa loaf recipes. I also found some vegan/soy free fake shredded cheese. Daiya is the brand, I think. It melts and isn't disgusting. I did it for a while and then eventually decided that I didn't care for it...but it helped get me through.
One of my cousins used colic calm with her daughter and said it helped. It does change the stool color and I didn't try it because my son had a bloody bum from diaper rash. Not fun.
I'm sorry if this is rambling and haphazard. Definitely try the lactation consultant. You need eat more, drink more water, and try to get some help. I didn't have any and it was really hard. And I wouldn't have tried the formula either...I just was so determined to breastfeed. Talking to a good IBCLC should also provide you with some good emotional support.
Good luck! You are doing a lot of right things. Hang in there. it will get better.