While there are many posts here about how to make better weapons, I'll join Pete Kirkham in answering the question in the title: how do police work without guns.
I come from a sizeable town in New Zealand, and when I was growing up, police officers were not allowed to carry guns. (And tazers weren't around yet either). And even now police still do not frequently carry firearms. In fact, I can't think of a single time I've seen a police officer in NZ with a weapon. This does not stop them from doing there job, because:
A police officer is not a hero who shoots the bad guys.
What does a police officer do? They direct traffic, they certify photos for driving licenses and passports, they make sure people are obeying the speed limit. None of that requires weapons of any sort. When encountering a police officer in New Zealand, it's just like meeting any other sort of person. Sure, he's in uniform - but if you need directions to the nearest gas station, or want have a question about the road rules, just ask a police officer. Sure, in NZ there is a special "Armed Offenders Squad" to deal with the cases where you need a highly trained military force to deal with a situation, but a police officer is just a guy who can help you out if you need it.
When I went to Europe for the first time, and saw police officers carrying pistols, they broke that impression of the police immediately. Wearing a gun immediately separates the police from the civilians. They're no longer friendlies who can offer advice and help you out, they now become part of an enforcement agency who can use force to ensure citizens to do what the government says.
If your police is overly armed, there is very little difference between a police force and an occupying army. Neither results in a safe-feeling society.
The police do have methods for dealing with unruly people without firearms. From batons and riot shields to high-power-water-guns, police have trained in non-violent ways to keep the peace. The first step is talking to the people involved - most people in society are honest when directly confronted. Pulling out a weapon to kill someone is a sign that all other methods have failed.
I will also add that your situation works both ways. If normal clothing stops a bullet, then your "bad guys" will also struggle to kill anyone.