My adrenals are weak right now - low cortisol and I can't handle stress or any exercise beyond basic walking - even walking more than 25 minutes will make me feel quite tired. Also problem falling asleep at night, can't sleep until about 2 or 3 am.

Anyways, I am very interested in following in Unyquity's recommendations along with Dr. Schulze 's teachings for healing. Uny recommends taking a tbsp of ashwagandha 3x a day to heal the adrenals.

That is a real hefty dose.  I generally recommend about a half a teaspoon three times daily, double the dose in extreme cases.

I am personally wondering what the difference is between that and American ginseng. I have a bottle of grounded roots of the ginseng (very expensive) and am wondering if I could use that.

I understand that they are both adaptogens, but is one of them more stimulating? One of them better for the adrenals? Whenever I take the ginseng I feel more awake, but am not sure if will cause any sleeping problems to get worse

Panax ginsengs are stimulatory while ashwagandha is calming:

http://curezone.org/forums/fm.asp?i=1466802#i

I would not rely on a single supplement for adrenal dysfunction though.  Vitamin C and pantothenic acid are the most important nutrients needed by the adrenals.  And a mix of adaptogenic herbs will work more effectively that an individual adaptogens.  Some of the best are schisandra berry, Chinese licorice root, astragalus, jiaogulan and ashwagandha:

http://curezone.org/forums/fm.asp?i=1527351#i