DNS security
According to Linode, "Our DNS platform is secure, especially when communicating over the private network."
I'm not sure how to assess how secure is "secure". The "especially" part of that sentence makes me wonder what aspect of it is less secure than optimally secure.
Using IP addresses is not a good alternative because the nature of the application would make it hard to migrate to new IP addresses at another hosting company if circumstances dictated a need to migrate.
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DNSSEC
For a more realistic answer
There is nothing you can do about your end users' recursive nameservers other than transitioning to DNSSEC and hope they've got secure connections to nameservers that support and validate DNSSEC. This is not common.