Can I pause the campaign? If I pause it, will it affect my stats or alter my bid positions or adversely affect anything in any way when I make it go live again after propagation?
How much is the URL changing? If it is just the location within the same site, then you can create a permanent redirect in your htaccess file to direct any requests to the new page. This way you don't actually have to make a change to the ad.
If the actual site name is changing, then you have more problems. You might try a redirect from the old site to the new one, but that will probably get pulled down next time the ad is reviewed. You might try to create a new ad pointing to the new site in the same ad group as the other one and do the redirect on the url of the old ad. The old and new ad will alternate, and hopefully by the time the old ad gets pulled for the URL not matching the destination site, the new ad will have enough history to not cost you too much extra. You can always delete the old ad when you decide the new one is performing well.
As far as pausing, that does not seem to effect the stats negatively. I do it occasionally if I know something might be going on on my site such as maintainace or an outage that looks like it will last more than a few minutes.