I agree with Mr. Subjunctive here. The dark spots are normal and the dried "leaves" look like the sheaths from which the stalks emerge. If they are dry and papery, just remove them.

Your plant doesn't look poor at all to me. If you are the type who is inclined to "fuss" with your new plants, resist the urge with your ZZ. These are very tough. Let it dry before watering, keep it in moderate to bright light, leave it pretty much alone, and it will do well for you.

I was also thinking about those sheaths, of course, now that I wanted to see one, none of the 4 separate ZZ plants I'm currently growing have one at the moment.

Looking at the leaves & stems in the pix, I thought they looked good & healthy & feel that even more so after Mr. Sub's comments.

On the other hand, if I thought they looked bad & had only had them one week, I'd take them back to point of purchase & discuss w/ vendor.

Thank you both very much. I was worried it was dying on me because I hadn't noticed the brown/black spots before. It seemed as if they had appeared overnight. I will let it be and hope it continues to do well. :-)

Thank you for asking that question. Mine has the same black spots. I really like that exotic plant and I had it sitting close to the patio doors. The deck is covered with no sun at all and it didn't grow. I moved it about 12 feet from the picture window which does put sun in the room. In that setting it put up 4 new shoots. All I read about house plants is that most rest in the winter. Almost all of mine are growing quickly and putting up new shoots, especially my mother in law tongues.

The irregular black splotches on ZZ's stems are normal. The dried sheaths down at soil level are also quite normal. If you purchased a new plant and it was potted a bit too deep where it was grown , the lowermost leaves that are partially buried would also dry up and turn brown.