As a trained EMT and Outdoor Emergency Care instructor with National Ski Patrol what you are experiencing is High Altitude Sickness. This as you have stated can occur at 6500 Ft. Treatment is apply oxygen and get your a.... to lower altitude, 1000 ft at a time until you feel better.
Now, how to prepare for future ascents. Get in shape!!!! Start walking. You don't need to jog. Your goal is 5 miles at 10 minutes per mile 3 - 5 times a week. Just good cardiovascular execsize. Or if time is no object go to 4000 ft for 1 week and increase altitude by 1000 ft each week. You will need to do some cardio exersize during this week to prepare yourself for the next increase anyway so, you may as well do it at home.
I'm going on 54. I recently went from my home, at sea level where I lived all my life, to Colorado for work. I took a weekend ski trip at +10,000 feet with no problem other than getting short winded when I climbed an additional 500 feet for some untracked powder. I am physically active, but don't consider myself to be in real good shape, just fair.
Good luck, start preparing now. It will be worth it
PS: High altitude sickness is not serious. However it's follow on HACE and HAPE, High Altitude Cerebral Edema and High Altitude Pulminary Edema can be. Heed the warning signs, get in shape, take it easy and if the headaches continue go to lower altitude. I've had friends die from HAPE.