I know the idea of cutting food out of your diet could be upsetting, but try not to look at it that way. Think of substituting. When I found out that my nursing son was allergic to both milk and eggs (and I'm vegetarian to start with), I had to get really creative--especially without wheat and soy, which are staples for vegetarians. I was just as healthy then as I am now that he's weaned and can eat everything. But I stayed vegan.
I'm telling you this for two reasons. One, to encourage you that you don't have to worry about cutting out food groups if you can find good substitutes for them. Nuts are a great idea (like one other poster mentioned). Good fat can help to "grease" things a bit. Try making half almond, half cashew milk. It's got tons of good fat, is very satisfying, and you can flavor it however you want (try making chocolate milk!).
The other reason is that I know exactly what you're dealing with. Been there. And I think I can safely say since I have been vegan, the problem has disappeared. Not just sort of. It's completely gone. Cutting out dairy--especially cheese--might make the difference for you. The protein in milk is harder to digest than meat, so maybe you can just cut out milk and get your calcium other ways, and not gave to give up meat.
Hope this helps! I know how miserable it is... but I'm sure you will find the solution eventually. :)