I have had severe, and I mean severe, indigestion. Your comment of "radiating to the back" makes me think gallstones and you might ask your midwife about that. They're not entirely uncommon in pregnancy!
That being said you are NOT alone. I do get very severe indigestion. Hours of pressure in the middle of my upper abdomen, laying down makes it worse. I only found relief (and it still hurt, just not as much) by curling while sitting up. Even then, it was hours of pain before I finally vomited stomach acid and it eventually eased to painful, but bearable, indigestion pains. It was 2-3 times a month at first, starting around 20 weeks, usually triggered by acidic foods (spaghetti was my first known trigger), but by the end of my pregnancy it was a nightly occurrence whether I ate or not. Unfortunately, my doctor dismissed all of my concerns and just told me to take Tums (which I did, but it did NOT help at ALL). I even self-medicated with Prevacid, which also did not help in the end. I lost 4lbs in a week from not being able to eat, and finally went to the ER (who sent me up to L&D since I was 38 weeks) to get some help. Ended up in labor, though, and never got an actual diagnosis.
I had it with my first pregnancy, but not my second, but had an "attack" at only 14 weeks this time around. I told my doctor and he put me on Pepcid. It has helped, but I'm still getting heartburn, so I have the feeling I might still have issues in the 3rd trimester, but at least I have a doctor who will listen this time! Even if it's just indigestion in the end, you are NOT alone. It can suck, but, actually, labor's not much worse (if it's the same) and at least you get a baby at the end of that, so it's worth it.