Not in your room silly. Take towels outside, spray them out amd then let them dry.
Rags are rags Tahlz. Ya don't need to buy any. I use old t-shirts and like Kamdon, sometimes towels. After the rags and towels go through the wash ... you'd never know the difference.
Anytime I'm cleaning up any spills, it a lot of water is included, I get my most/best absorbent towels and use those, then wash them with some bleach.
IDK if you have it where you are,but Odoban works really well for the smell.
Could you get puppy pee pads and set up a controlled area he can relieve himself, or is it completely out of his control?