Hi Postmeno3,
You said, "The 'sweats,' I feel, are different to 'night sweats' in that they only happen if and when I wake, once awake. They do not wake me."
Another cause for the sweating could be low blood sugar in the morning (nocturnal hypoglycemia). While sleeping at night, your blood glucose level may be falling too low. Adrenaline is produced in response to the declining blood sugar level which causes sweating.
Depending on the severity, other symptoms include nightmares or sleep disturbances, headache, lightheadedness or dizziness, anxiety, pounding heart, dry mouth, tingling or numbness of the mouth, shaking, hunger, blurred vision.