I'm affiliated with two universities: do I need ethics approval from both?

Yes.

Or at the very least, check with both Universities' ethics boards.

I know of one instance of a post graduate student who in a similar situation got clearance from one university, but not the other, and was forced to discard their results.

The purpose of IRB approval is for four things. The first is to make sure research is being done ethically and responsibly. Except when used in a circular definition, research does not need IRB approval to be conducted in an ethical manner. It definitely does not need multiple IRBs to approve it. The second is funders may not fund research that is not approved. They would only want approval from the institution where the funding is being given to. To publish research, it general needs IRB approval, but again, they do not care where it comes from. From both the publisher and funder point of view, IRB approval is their proof that the research was conducted in an ethical manner.

The final reason is so you are not liable. IRB approval, at least at the universities I am familiar with, means that the university takes on the liability as long as the protocol is approved. As place that may get sued,more any place that you will want help from if you get sued, will likely need to approve the research. Many IRBs have a light touch review for studies that have been reviewed by another university's IRB.

In the US, you generally need the approval of an Institutional Review Board (IRB) if your work was funded by the federal government, and you would have to demonstrate that approval to publish in any venue that required it. There are probably many other institutions and laws in the US that require IRB approval for this kind of research, but most of the issue is driven by the federal government's attachment of the process to its research funding. If this were happening here, either one should be sufficient. That being said, if the work is mostly related to your postdoc, I would be most concerned about the requirements of the funding agency or university that funds it. An adjunct position in the US would imply mostly a teaching role to me which would be less concerned about your research work. Your situation may be different.