Changing affiliation on publication

I would argue that it's a question of resources—if you have used resources (equipment, personnel, computers, financial) of both institutions in executing the paper before its initial submission, then you should list both institutes as supporting the work.

On the other hand, if the core of the work—both research and writing—were done at the old institution, then the new institution should only be listed as a "present affiliation," rather than as a "co-sponsor" of the work.

It is not necessarily clear cut. As a principle I would say that the affiliation should be where you have done most of your work. It is, however, also possible to use two affiliations and set your new location as "present address:" (or affilaition). After all, the affiliation serves two purposes. For a university (department) it means something becomes associated to them. For you, though, it means people can find you if they have, for example, questions. So, using the double affiliation, if it is possible, might be a good idea in the grey zone. Otherwise, I would stick with "where you did most of the work", however you wish to define it.

Your name and affiliation are listed on publications to identify you and to facilitate contact. Thus, I believe your new institution should be listed, rather than your old. Your old institution can be credited in the acknowledgements. Given that any affiliation listed might become outdated, you can also list your personal email address and your personal website URL. (A personal domain name and web hosting are very cheap.)