I too will sometimes get breathless after walking or it will feel like something heavy is sitting on my chest, especially when it's humid outside. I was diagnosed with GERD about a month ago and when I told my doctor about my breathing problems, which started happening a few months after I started developing a more acid feeling coming up, he didn't seem concerned at all. I've had my lungs and heart checked, everything was fine. I even posted about this a few days ago, but no one replied. I did some digging on this forum though and I found a few people talking about the same issue and that sometimes the acid fumes or even the acid can leak into your lungs, which can cause damage, I researched it more, and sometimes, not always (so don't worry), it can cause lung cancer, but that's if it gets serious enough. It made sense to me, acid isn't good, unless it stays in your stomach.
Your symptoms are like me, I get a weird feeling from my stomach to my throat, and sometimes I will even get mild chest pains, which my doctor said was usual with some patients with GERD. If you are worried, make an appointment with your doctor, and talk to him/her about what you're feeling and if it could be GERD. There's millions of people that have it, so even if you do, don't feel like you're the only one.