Sleepy tired vs physically tired

Solution 1:

I think participles might help give the sense that something (e.g. physical exertion) has “worn you out” beyond the normal process of becoming sleepy. For example, wearied (as opposed to simply weary).

I think some colloquial constructions that use a passive form might also help give the sense of what you want, like these:

• done-in: worn out; exhausted; used up
• drained: to be deprived of strength
• sapped: to drain the [every; vitality] from
• spent: used up; consumed
• beat: exhausted; worn out

Solution 2:

worn-out (from Wiktionary, 2019):

Exhausted or fatigued from exertion.

I think exhausted or depleted could also work when provided with more context.

The thing is sleep is a very complex state due to circadian and homeostatic processes.

Circadian: someone who is full of energy but has a strict sleeping pattern can still be sleepy at bed time.

Homeostatic: someone who is depleted of resources can still be wakeful or restless due to changes in hormonal processes such as adrenaline rushes from bodily stress and blocking adenosine from binding to its receptors when ingesting caffeine.