Need clarification on switching CiviCRM hosting from a host/consultant
Yes, you can move your Civi installation to new hosting (though it is recommended to use a knowledgeable/experienced provider to do so; and you sound like you are on board with that idea). A good place to start is with one of the providers listed here: https://civicrm.org/partners-contributors

Here is some background on those providers listed and some tips on selecting a new provider: https://civicrm.org/about-service-providers