I replicated our DNS on DNSMadeEasy, which I've used in the past on a very high-traffic website. They have an incredible track record for uptime. And they're really cheap.
We manually exported our DNS configuration from DNSimple as a single file and imported it to DNSMadeEasy. Pretty easy transition, although we have to make future changes in two places.
I set 2 nameservers on the old host and 2 on the new, and am keeping that configuration for automatic failover in the future. The DNS propagated fairly quickly and our site was up and running for many users within an hour. Glad that's over with.