Yes! To using poorly - I use it & I like it. Reminds me of the Blur song "You're So Great":
Sad, drunk, and poorly
Sleeping really late
Sad, drunk, and poorly
Not feeling so great
Wandering lost in a town full of frowns
Sad, drunk, and poorly
Dogs digging up the ground...
Also, IME here, to say you're sick means you're in a vomiting kind of way, whereas poorly or unwell is more general like you could have a cold or whatnot.