Do draft products count towards the number of products my plan allows?

No. Your plan limits active (published) products. Drafts do not count.

On Products → Products, the badge next to the title shows your usage, for example "8 / 10 active". Hover over it to see your total number of products, drafts included.

When you reach the limit, you can still click + Add Product, set Status to Draft and click Save. To publish one, free up a slot first: open an older product, set its Status to Draft (or move it to Trash) and click Save. Then set the new product to Active. You can also upgrade from Settings → Account → Subscription.