So it seems to me that signs of stress would be high levels of both of these things.
Yes. Wikipedia claims  the same:
Stressors [...] activate the HPA axis, though via different pathways. [...] Stressors that are uncontrollable, threaten physical integrity, or involve trauma tend to have a high, flat diurnal profile of cortisol release (with lower-than-normal levels of cortisol in the morning and higher-than-normal levels in the evening) resulting in a high overall level of daily cortisol release. On the other hand, controllable stressors tend to produce higher-than-normal morning cortisol.
The same on a webpage of Society for Endocrinology :
In addition, in response to stress, extra cortisol is released to help the body to respond appropriately.
Yet, there are conditions in which stress can lower cortisol levels. One of them is psoriazis :
[...] patients with persistently high levels of stressors seem to have a specific psychophysiological profile of lowered cortisol levels and may be particularly vulnerable to the influence of stressors on their psoriasis.
And some psychiatric disorders :
Several stress-associated neuropsychiatric disorders, notably posttraumatic stress disorder and chronic pain and fatigue syndromes, paradoxically exhibit somewhat low plasma levels of the stress hormone cortisol.
- Wikipedia contributors, "Hypothalamic–pituitary–adrenal axis," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Hypothalamic%E2%80%93pituitary%E2%80%93adrenal_axis&oldid=619115237 (accessed August 7, 2014).
- Society for Endocrinology. Hormones. Cortisol (2013). Available from http://www.yourhormones.info/hormones/cortisol.aspx (accessed 07.08.2014)
- Evers AW, Verhoeven EW, Kraaimaat FW, de Jong EM, de Brouwer SJ, Schalkwijk J, Sweep FC, van de Kerkhof PC. How stress gets under the skin: cortisol and stress reactivity in psoriasis. Br. J. Dermatol. 2010 Nov;163(5):986-91. doi: 10.1111/j.1365-2133.2010.09984.x. PubMed PMID: 20716227.
- Yehuda R, Seckl J. Minireview: Stress-related psychiatric disorders with low cortisol levels: a metabolic hypothesis. Endocrinology. 2011 Dec;152(12):4496-503. doi: 10.1210/en.2011-1218. PubMed PMID: 21971152.