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Also, I'm already disappointed with the new hosting service and considering yet another move for my sites. Would this be a disaster at this point?
Your site could be completely offline and it shouldn't affect the update. You need to make sure that your site is accessible by Google during the deepcrawl.. that's when it's going to be important that they have new DNS information. If your site was crawled during the most recent one, then you shouldn't have a problem with the new index.
It's next month's update you should be worried about, if anything. I've heard from others that Google goes well over a month before updating their DNS cache, and then I've heard more recently that they are updating more frequently. By the time they crawl again, it'll probably be 3-4 weeks since your site was moved to the new host, and they may very well have updated their DNS cache.
Your hosting company (or whoever actually controls your DNS) tells how long that caching is SUPPOSED to last. A cache of 1 day is typical, and not that huge a problem.
The problem gets truly problematic when some ISP's (notable AOL) set their DNS servers to actually cache the record for a month.
The best I can say is good luck. When I worked at a hosting company we learned this the hard way when our upstream provider made us change IP addresses. Hope you're with a solid, reliable, well-connected hosting company now.
dwilson