I am recreating my website on another server. So my big concern is how to move all CRM data to new server/website. To do separately exporting/importing contacts, membership, activities etc. is not so much accurate way, because there is a possibility of missing something.
Also I can't dump whole database. Is there any possible method, how can I do this??
I think you are saying that you have a single database for CMS content on the old site and that you already have new CMS content in the database for the new site. I think you want to move the CiviCRM structure and data. If I have understood that correctly (it would be good if you clarified by editing the question) then you can do the following.
1) split the CMS and CiviCRM databases: KEEP A BACKUP! Create a copy of the combined database by dump, rename the original and then reload). In the original you delete the CiviCRM tables (with e.g. phpMyAdmin) and in the copy, delete the WP tables. You need to edit the civicrm.settings.php to change the name of the CRM database - it will have the same username when created this way. This last bit allows you to keep access to the old CiviCRM so you can check things.
2) on your new site, reinstall CiviCRM using the same version as the old site and the same database details. Now replace the CiviCRM database with the CiviCRM database from the old site. Have a look at the documentation first! https://docs.civicrm.org/sysadmin/en/latest/misc/switch-servers/
There is now this solution that may help some folk https://lab.civicrm.org/extensions/configitems
