I am not sure what you mean when you say you want to avoid liability “from our old namesake.”  Since the organization does not change on January 1, it will be liable for anything that it did before January 1 in its old name, regardless of what it may be calling itself at the moment.  Another organization that takes your old name after you give it up should not create liability for you, although you may have to spend a lot of time and money to clarify that any liability it created after taking your old name belongs to it and cannot be attributed to your organization.

It may make sense to retain your old domain names and redirect visitors to your new sites, at least for a while.  That will make it slightly more difficult for a new organization adopting your name to operate as if it were you.  But the basic principle is that liability belongs to the organization that created it, without regard to what the organization calls itself.