It has been a while since you posted this, but it took me years to get a diagnosis so you might benefit from my response.
I went through the exact same thing, right down to being kicked out of the gym. My GP did a resting EKG and pretty much called me crazy. He gave me permission to return, but when it kept happening, I went to a cardiologist. I had a 24 hour heart monitor, stress test and echocardiogram, and was told I was fine. They put me on a beta blocker and it didn't help at all. After that, they sent me to an electrophysiologist, which is a cardiology specialty that looks at heart rhythms.
The ep put me on a 3 week heart monitor and did a tilt table test, but I think that was overkill. After all that, he said my symptoms were common and classic. I don't know if it is what you have, but they said I have Vasovagal Syncope.
You know those Victorian women who fainted when they got excited or those people who pass out at the sight of blood? My body responds the same way to heat and exercise as theirs did to emotional triggers. Heat and exercise both cause a person's veins to dilate. Fair skinned and thin people are particularly sensitive to this (noting your avatar, I'd guess you are at least one of the two). In people with vasovagal syncope, the veins in the legs don't constrict enough to get the blood back up to the head. That causes a fainting, lightheaded, dizzy, or sleepy feeling as the blood pressure drops. Your body is going to get blood to your brain one way or another, so if you don't do it, your body will do it for you in the form of passing out. If you lie down with your feet elevated, it should pass without your losing consciousness. Some people lose consciousness so suddenly that they don't have time to lie down.
The good news is that most people eventually out-grow vasovagal syncope in middle to late middle ages. Another benefit is that it isn't life-threatening as it is what is called a self-correcting condition. If you get to the point of fainting, your body will correct the problem while you are passed out and you'll come-to just fine. If you're like me, you'll want to avoid the fainting in the first place, and there are some measures you can take to do so.
Ask your cardiologist if you would benefit from an increase in salt intake. Some cardiological issues would be hurt by salt, so please do ask. Salt causes fluid retention and it significantly helps in low blood pressure. My doctor told me that it didn't matter how much I was drinking, that I needed to double it. I had to eliminate all diuretics (soda, coffee, tea, etc) from my diet as much as possible, as they get rid of fluid. And I had to re-structure my day to exercise, shower, and do other physically demanding or hot tasks well after I woke up. Blood pressure and hydration levels tend to be low in the morning, and can be really problematic if you have hypostatic orthotension as well (blood pressure drops when you change positions, like getting out of bed or standing up after doing floor exercises). By giving myself more time to get going, and drinking plenty of water at least 30 minutes before a task, I've almost eliminated my symptoms and the rest I'm able to control by taking a break to lie down when I first notice the need.
If you are still having problems, I strongly encourage you to look into this relatively simple diagnosis.
Susie