I got a phone call telling me my iCloud account had been compromised and to call Apple Support.

Such calls are scams from criminals trying to steal your personal and financial information. Hang up on them, repeatedly if needed.

Avoid phishing emails, fake 'virus' alerts, phony support calls, and other scams - Apple Support

You got an email claiming to be from Apple; changing your password probably wasn't necessary but worth doing to be doubly sure. Just never respond to any email of this sort - the links will be crafted to look like Apple but aren't.