Do we just say, I’ve had enough sleep, or I’m full of sleep or something else?
You could say "I am well-rested".
If you want to refer to last night, you could also say "I had a good night's sleep" to indicate the amount of sleep you got was sufficient to plentyful.
You can simply say that you had a good (night's) sleep:
a night in which one sleeps well.
Did you have a good night's sleep? I need a good night's sleep to be at my best in the morning.
One more way to say enough sleep is salubrious sleep. Salubrious sleep is healthy sleep. You feel you had an enough sleep. It gives you a feeling of contentment.
Salubrious Oxford dictionaries
Citation from the books:
Healthy Living in Late Renaissance Italy By Sandra Cavallo, Tessa Storey
They made for more salubrious sleep according to the principles of healthy life promoted in the advice literature...
The Critical Review: Or, Annals of Literature, Volume 21 edited by Tobias George Smollett
We, indeed, consider dreaming to be only an interruption of salubrious sleep.