I don't think you're nuts at all, but I would think very hard on how you would approach them and what you're going to actually say. You can't imply that the current site is an embarrassment to the community, the very person you speak with may be the one responsible for the mess. What I would do is be totally honest, leaving out any comments about the poor layout of their current site, and just tell them you are new to the area and felt their site could use some updating and would like to show them a sample of what you could do for their website. And don't ask the first person you meet up with when you walk in the door. Ask to see the person who deals with the site's information. I also wouldn't blurt out that you'd do it for free, wait to see if they're willing to pay first. It might be within their budget to have some work done on it. Your own website should demonstrate your design abilities, so by all means have them get to their sample through a link located on a secondary page of your website.
Good luck! I'm new to my area as well and am considering doing the same. Last time I did this I ended up on a Chamber of Commerce committee and they hired a huge company to do their site. I'll not fall into that hole again! ROFL