Thanks for the comforting thoughts and group hugs, everyone... 
It looks like there might be a light at the end of the tunnel. I'm still down right now, but I finally got a promise of progress today. Turns out the customer support may not be bad after all. They just suck at communication.
When I noticed the initial slowdown Thursday morning, I called their support line and explained what I was seeing to one of the guys there. He checked his console and said he did see a problem, conferenced the call with one of their system administrators who said he would check into it immediately and that slow response times like that were usually resolved within a half hour or so. I figured that sounded good enough, we hung up, and I went to bed. I woke up that night to find my site was not responding at all and there was no email from anyone telling me what the status was.
So, I went to their web site and opened a ticket, explaining what I had been told in the morning and requesting a status update. Early the next morning I got a short email that just said "The server you are on is under a denial of service attack".
Ok... That tells me what the problem is, but it doesn't tell me what you're doing about it... So, I reopen my ticket (another gripe I have -- when they reply to a ticket message, the ticket is automatically closed. It shouldn't be closed until the problem is resolved!!!) and add a request for a status update. First thing the next morning I get another short email saying only "The attack continues".
Ok, now I'm furious! I KNOW the attack continues! I can see that when I try to access my site and it just times out! I was getting the distinct impression that they weren't doing a damn thing about it, but just waiting for the attackers to get tired of their game and go away. In the meantime, my site has been down for two days. My short term goal is for this site to become a small business. If this had happened while it was a source of income for me, possibly even my sole source of income, I'd be consulting an attorney at this point.
I reopen the ticket and add another line, trying my absolute best to not go on a long rant, riddled with foul language, and limit myself to one line (if they can be terse, why not me?): "But are you doing ANYTHING to stop the attack?!?"
First thing this morning I got another email. This time it was actually TWO sentences! Still pretty short, but at least they finally told me what is going on. They have shut the affected servers down completely and are in the process of migrating everything over to a completely new bank of IP addresses, and hope to have everything back to normal within the next 12 hours, and to let them know if my site is not up within that time.
So, if I'm back up and running in the promised time, I'll concede that they do in fact have good customer support as I was promised when I was referred to them. They seriously need to work on their communication, though. I have the email address of the company's CEO, and will probably be sending him a note with a suggestion or two once this is all over with.
I work in IT and man the help desk during my shift. A lack of communication like that during a major system outage would have me in a heap o' trouble!