I've always had puffy lower eyelids in the morning, but now my upper eyelids get puffy too. Is this normal?

Maybe. Puffiness of the lower and upper lids can result from fluid retention, sometimes worse in the morning due to lying horizontally while sleeping. When you stand up, gravity pulls the fluid down. The most common cause of "puffy" lower lids is fat prolapsing forward mimicking swelling. The upper lid skin can stretch and become redundant with age appearing swollen.