Switching Host

[Josh Knepfle]

Hello,

I am switching hosting facilities for our sites.  I'm having a little
trouble, and I was wondering if someone could weigh in on the issues
I'm having.

Here is the procedure I have followed:

1. Set up DNS entries in our New host's DNS servers to point to our
old (current) IP address, with a short TTL of 60
2. Changed the DNS servers in the domain name registrations to the DNS
servers at the new facility
3. Wait the CURRENT TTL (one day) for our domains for the changes to
propogate.

When we switch the servers to the new facility, we plan to change the
IP address of the domain names to the new IP's and hope that everyone
will go to the new IP within 60 seconds (the new TTL)

The issue that I'm currently having is that one of the DNS servers
from the ISP in our office is still using the old DNS server from our
domain name registration to get the IP for our domain.  The other is
fine.  I can fix that for myself, no problem.  The issue I'm facing
is, what if other ISPs are doing the same, even though it would seem
that they shouldn't?  I called our ISP and they said that as long as
the old hosting facility's DNS servers are claiming to be the
authority for our DNS, their DNS servers are going to continue to look
to them for our IP address.  That doesn't seem to make logical sense
to me.  I can think of a case where a company wished to switch
providers, but where the old provider never removed the entries in
their DNS servers, so that some segment of the population continued to
look to their DNS servers for the IP addresses.  In that case, a
malicious provider could force a client to never be allowed to leave.

Ok...assuming that that IS the case, and the rules of "what should be"
have been broken...what can I do?  My thinking is that the ISP in my
office has their DNS servers set up improperly since one of the DNS
servers I'm using here in the office is correct and the other is not. 
Any thoughts?