Did the filmmaker's forget Forensics 101? SPOILER!
Are we to assume that the filmmakers forgot about fingerprints, or are we to assume that they want us to think that Paul did? Paul's fingerprints are on the phone and credit card. Police aren't just going to accept what the scene, and the entire episode, appear to be without any investigating. Sheesh!
Not only that, but it's highly likely that they will be able to tell that Dean didn't shoot himself.
Time wounds all heels.