Hello ******, I would think that a hemoglobin of 6.5 his really low. I found out I was anemic 2 years ago, I had the same symptoms as you but just shrugged it off thinking it was because I was overweight. I especially can relate to the feeling in your head when you stand up or when you just exert yourself a little bit and your shortness of breathe. I became anemic because of heavy periods. And I never really thought I had heavy periods, even when my doctor asked me if I have heavy periods, I was like no I think there basically normal, so you see even though you think you have a normal period it could likely be heavy enough to make you anemic. I still take 130mgs of iron daily because my period is still heavy. And I did start to feel better almost right after starting the iron pill.
Anyway take care&hope you start to feel better real soon,