How do I specify my affiliation/contact information when I changed institutions and fields?
If you have the opportunity to be clear (that is, some limited online form doesn't prevent it) explain the transition "Jon Barker, formerly of Acme Corp, now with Zypppy Inc. JB@Zypppy.com". Or something similar. 

The reason for the email is so that people can contact you, so use the current one even if it seems weird. Using an obsolete mail does no one any good. 

If the former employer/institution contributed funds to the project it is usually good etiquette to indicate that you were associated with them then. But for identification purposes, your current affiliation is more valuable. But if it is a paper with multiple authors, some still at Acme, then it is less important to list an old affiliation (my opinion only). 

There may even be reasons that you must not imply an affiliation that no longer exists, but that is fairly rare. If you can only give a single affiliation, it is probably best in most cases to list your current one. I suspect such a rule has many possible exceptions, but I'd suggest thinking about why you think you have an exceptional case.