Well, you're already in the midst of downtime, so you have a head start.
Aside from that, focus on two concepts: TTL and overlap.
1) On your old host, lower the TTL of all your records to something quite low, like 30 seconds. This will increase the burden on your nameservers, since records will only be cached that long, but it will make you more nimble as you make substantial changes.
2) Migrate your records over to the new provider. This can be a bit tedious for more complex zones, but rather straight-forward for many. Change your nameserver on your local machine to point to your new DNS host, just to test in a "real-world" scenario.
Then go to your registrar and flip the switch.
Switching DNS providers is much less prone to downtime than changing web hosts, since the records themselves aren't changing much -- just where to find them.
If you're switching registrars, the principal is similar, setting the TTL quite low during the transition to help you make changes more quickly should something go wrong.
Imo, having done this since the mid 90's, you don't have to mess with TTL since you aren't changing any of the records. And having someone do that is an additional thing to mess with that adds unnecessary steps.
TTL would be necessary if you are going from one IP to another or a different MX server etc. though.
Altering TTL won't do anything for you unless you do it before you start to make changes and far enough in advance that the current TTL will expire and servers will pick up the new TTL. Then when you alter DNS records the alterations should propagate much faster.
If you're literally just changing DNS providers why would there be any down time - the record showing the IP where to find your website just gets grabbed from a different location, if a stale record is used it's still right.
Aside from that, focus on two concepts: TTL and overlap.
1) On your old host, lower the TTL of all your records to something quite low, like 30 seconds. This will increase the burden on your nameservers, since records will only be cached that long, but it will make you more nimble as you make substantial changes.
2) Migrate your records over to the new provider. This can be a bit tedious for more complex zones, but rather straight-forward for many. Change your nameserver on your local machine to point to your new DNS host, just to test in a "real-world" scenario.
Then go to your registrar and flip the switch.
Switching DNS providers is much less prone to downtime than changing web hosts, since the records themselves aren't changing much -- just where to find them.
If you're switching registrars, the principal is similar, setting the TTL quite low during the transition to help you make changes more quickly should something go wrong.