Why do I have these? There are no links from example-old.com to example-new.com.

I can answer this part, because it came up for me when I moved sites near the end of last year. In my case, I moved 6 of 8 directories from oldsite to newsite, leaving 2 at oldsite. For months and months afterward, gwt listed vast numbers of links between oldsite and newsite (in both directions) "via this intermediate link". In reality these were internal links within either oldsite or newsite, including the many links within directories that were on the same domain. Because not everything gets re-indexed at the same time, you'd have

example.com/directory/pagename
LINK TO
example.com/otherdirectory/otherpage

but by the time the computer got done with it, it looked like

example.org/directory/pagename
REDIRECT TO
example.com/otherdirectory/otherpage

That's where the "via this intermediate link" comes from.

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On your main question, it may be more useful to think in terms of what you want the user experience to be. You certainly don't want to give up any old domain names, because you're looking at $10/year to keep them alive vs. at least $1000 to buy them back from a dragon. You also don't want the DNS for more than one domain to point to the same physical location, because duplicate content. Sure you can do stuff with "canonical", but anything involving the word "canonical" is generally a second choice.

So the question is whether you want a user requesting
example.com/some-bookmarked-or-linked-page
to end up
-- on a browser error page saying "I can't find the server"
-- on a browser error page saying "It seems to be a valid domain, but I can't open the page"
-- on a single parked page saying "This site doesn't exist any more, so you want example.org instead"
-- redirected to the same page at example.org, assuming a one-for-one correspondence

Then, once you've figured out what you want to have happen, you change the registrar and/or DNS records accordingly.