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The problem is, my domains are using Registrar A, and the registrant is Host A (I use anonymous registration, so the old host is the registrant). If I transfer the domain registrations to the new host, they will use another Registrar, so my domains will be transferred to Registrar B, and the registrant will become Host B. However, the site content will remain the same. Will Google and other search engines consider this as a change of ownership and affect the ranking of my sites?
As for transferring the domains, this is a good opportunity to separate your domain registration from your hosting provider - leaving all your eggs in the same basket is bad practice and can limit your options for moving between different hosting providers. Choose a good registrar independent from either your old or new hosts.