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If we move it off one domain and onto the other, and Google crawls the new site before the old one, then it will see that both sites have the same content, regardless of the fact the content was removed from the old site.
We don't want to bring the content offline even for a couple of days. Any good tips?
Here is a thread on a very similar situation:
Changing domain names [webmasterworld.com]
Hope this helps,
Regards
William
Being on shared hosting doesn't enter in to it directly. The question is simply whether your old-domain host supports server redirection (as opposed to scripted or meta-refresh redirection). If you are on Apache server, the question is, "Can I use mod_alias or mod_rewrite to do a 301-Moved Permanently redirect?" On MS servers, I can't give you the specific question, but generically, "Can I don a 301 redirect to another domain?" is close enough.
So research those questions indirectly, or ask your old-domain host directly. (Note that it makes no difference if the new and old domains are hosted by the same company, or even on the same shared server, the question is the same.)
All the rest is just implementation details, and it's not so hard to do.
HTH,
Jim