I don't see how paper bags aren't better. Both plastic and paper require processing. The difference is that paper isn't toxic when degraded, plastic is.

Both are energy-intensive, paper bags degrade, but also they tear, soak and shred much faster, so you need more of them. And then the energy used in manufacturing them makes them not very eco-friendly.

As I said, get a cotton bag and sue it forever, if you're concerned about the environment.