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Any suggestions on how to prevent this?
Should I dump the old site totally? I have kept it up because Google is still directing traffic to it. When I submitted the sites to Google I mentione this is a new domain of an old site.
I'm not sure if this is just a change in Google ranking or a problem with keeping an old site online.
The way you're doing it you are actually telling Google's bots and algo two things:
1. Your old page still has the same address. For the time being the content has changed.
2. Here is a new page. It has the same content that the old (and still existing) page had short time ago.
This is a rather problematic message. Especially if the old page still has lots of incoming links from other websites and the new page has very few.
I suspect that for the time being Google regard your pages as duplicate content and simply choose the old pages as the ones to show in SERPs. Eventually the algo will figure out that the old content has gone permanently from those pages, but by keeping them working you keep Google's doubt alive.
> When I submitted the sites to Google I mentione this is a new domain of an old site.
No need to submit anything to Google. And no need to tell them about your way of doing things. They do not have the time to deal with situations as undramatic as yours. Your websites are most likely not banned or penalised but just caught in a temporary problem that eventually will solve itself.