move part of table postgresql to another database

I'd use the \copy command. The copy is described here https://www.postgresql.org/docs/current/sql-copy.html. The \copy is available inside psql. The difference is that copy assumes the path to a file is on the database machine, while\copy assumes it's on the machine where the psql runs.

You can use \copy like this (this is an example from the above page):

\COPY (SELECT * FROM country WHERE country_name LIKE 'A%') 
TO 'country_data';

This way you have all the rows written to this file. Then you can load it to another database also using copy:

\COPY country FROM 'country_data';

Then you can delete the data with:

DELETE FROM country WHERE country_name LIKE 'A%';

You could also make copy and delete in one query. However, in this case I'd rather carefully check if the data in the new database is fine before deleting it.