You might consider getting a CCW permit, and then doing your walking without your gun (I think it would be very difficult to research all the local laws that might affect carry given that you're walking and have less control over what property you might cross).

I think it would put a cop at ease to hear, "Hello officer, I want to let you know that I have a CCW permit, but I am unarmed." This should let the police know that you're a good guy, because bad guys don't go through the trouble and expense of going through an FBI and state police background check on their own. Note: CCW permits are for firearms, they do NOT apply to knives (at least not here in Colorado).