Is it possible to host one TXT record for a domain on a different server, and leave all of the other DNS information intact on the first server?

Solution 1:

It depends on the exact details. In the cases where a CNAME is valid, you could add a CNAME which you could use to have your records exist in another zone hosted somewhere else.

# example.org zone (on bad ISP)
foo.example.org. IN CNAME foo.delegated.example.org.

delegated.example.org. IN NS goodisp-ns1.whatever

# delegated.example.org has been delegated to other ISP
foo.delegated.example.org. IN TXT "blah blah blah ..."

But a CNAME wouldn't be valid in all places.

For example you couldn't have it at the root of the zone.

Also, it may be possible that some software that checks the TXT record won't follow a CNAME.