Can stress and arousal be independent?
Stress response has 2 main components:


Quick response, within minutes, is by the Sympathomedullary Pathway (SAM): hypothalamus > sympathetic nervous system > release of adrenaline and noradrenaline from the adrenal medulla > stimulation of the heart, dilation of the muscle arteries, constriction of the gut and skin arteries, glcogenolysis (the breakdown of glycogen into glucose) > more glucose available as a fuel
Delayed response, within hours, is by the The Hypothalamic Pituitary-Adrenal (HPA) System: hypothalamus > pituitary gland > ACTH > release of cortisol from the adrenal cortex > gluconeogenesis (formation of glucose in the body from other substances) > more glucose available as a fuel


Cortisol secretion from the adrenal cortex can be stimulated by things other than stress, for example, hypoglycemia, but hypoglycemia will simultaneously activate the sympathetic nervous system, which will trigger the release of adrenaline (Neuroendocrine Response to Hypoglycemia).

Some cortisol is also secreted along with aldosterone in response to hyperkalemia or hyponatremia (Vivo.colostate.edu), for example in SIADH (A Case of Transient Hypercortisolism Simultaneously Occurring With the Syndrome of Inappropriate Antidiuretic Hormone Secretion Induced by Olanzapine).

Cortisol can be also increased without stimulation of the sympathetic system in Cushing syndrome due to a pituitary adenoma or, rarely, other tumors that secrete ACTH or in adrenal adenoma, which secretes cortisol.