Is there a way to get stale DNS records from SSM or AWS Trusted Advisor?

What is a "stale DNS record"? What I think you're asking is "how can I tell if a DNS record is pointing to an AWS resource that doesn't exist anymore".

The short answer to your question is no, there isn't an easy way to do that. And you should also consider stale CloudFront origin configurations.

If so: The resource records of interest will be A, CNAME, MX and ALIAS. Perhaps NS too. There's a good (external) post on this topic but it only does half the job. If you have a multi-account environment then you will need to create an inventory of resources (IP addresses, load balancers, etc.) that are available in all of those accounts and then check the Route 53 entries against them.

For CloudFront, take a look at this post.