I originally had a reseller account, but now have decided to go with a dedicated server as I've grown past what a reseller account can offer. I've read all the instructions on switching DNS over with minimal downtime but from what I understand any of the steps that would allow me to keep the same nameservers and change with minimal downtime require root access to the old server - something I don't have.

Is there any way I can do this without a day or more of downtime?

Well, if you can have the person who has root access change your records AND the ttl values this can be done.

Another way would be to get all your sites setup on your new server, and then switch the registrar information. You might have some downtime if you are generating dynamic content, and mail could be problematic for a bit, but this would not require root access on the old server.