Use the same cast iron skillet or buy a new one for desserts?
Tl;dr You can use your cast iron for anything if it is seasoned and cleaned well.

I don't have a ribbed pan and it seems that would be harder to clean so I will speak only to the smooth.

My cast iron pan is reasonably well seasoned. It could be better but I only get a little sticking.

Scraping it well and scrubbing with kosher salt and a paper towel cleans it spotless from almost anything. Occasionally I have to rinse it out which doesn't damage the seasoning at all.

When it is cleaned there are no lingering flavors from previous dishes. I often cook meat, onions, desserts, and cornbread with no problems.

Cornbread has a delightfully different and deliciously rustic texture when cooked in hot cast iron.