"But I would also strongly encourage you to get a cardiology work up. No, they don't need to bring you to altitude, but an EKG, echocardiogram, holter monitoring, stress test, etc can bring out activity-related issues. There are some very serious congenital cardiac conditions that don't show up until the heart is under stress (which would be exacerbated at altitude) and you would be smart to make sure this isn't the case."
Justin I know how you feal, because I have been there. In scouts I would frequently get sick on the first day of a hike but then rapidly get better and would be fine for the remainder of the hike. Since this was in the cascade mountains in Washington state altitudes were generally lower.
After collage I started day hiking in Yosemite and then backpacking. At lower elevations I would get sick but then quickly get over it. However once I started increasing the altitude it got worse. Sometime I was fine but other times it was bad on one trip when I was above 8,000feet I was constantly nauseous and it didn't clear until I dropped below 8,000 3 days later It was a very difficult hike. I talked to my doctor a few times about it and had some tests but nothing was found. I tried diamox but it just made things worse. I also tried supplements, more exercise and diet changes out nothing seemed to work. Sometimes I was fine. Othertimes I had to abort trips.
In the end I stopped talking to my doctors about (a big mistake) it and just stopped trying hiking at altitude.
4 years ago I was rushed to the hospital with a stroke. It didn't last long and left no long term problems. It was quickly followed by subsequent smaller ones. The doctors stopped the strokes with blood thinners and then started looking for the cause. 3 days and several tests later. I was diagnosed with a Atrial Septa l defect. A common birth defect. About 5% of the population has it but Most don't show any symptoms (or at least ones doctors would quickly recognize). I did either unless I stressed my body significantly. which means I had no symptoms at the doctors office, only on the trail.
Basically as long as my pulse was not reasonably low I was ok but as it increased my heart became increasingly ineffective at delivery oxygen to my body. Altitude just made it worse. My stomach would often not work well if at all and I was frequently light headed at altitude. Increasing food consumption and only made things worse and when you are not feeling well water consumption goes way up. Long Rests also didn't help me adapt.
After the defect was correct (the operation only lasted an hour and left no scar). I initially had a great deal of strength after the surgury and altitude no longer had any significant effect on me (at least for a while, it didn't last). I am now getting back into hiking and am really enjoying it.
See a cardiologist and your regular doctor and continuously work with them to figure out the problem.