So. In the end it was enough to change the address in the database. In the table wp_options there are the two entries pointing to the url that were set. By modifying those it should work. Unfortunately Chrome does some mumbo jumbo with redirects etc. and it kept pointing it to the old url. Testing it on another browser and/or clearing the cache did the trick.
Thanks to @Mayeenul Islam that pointed it out! He also suggested deleting .htaccess but I didn’t trust myself enough to do it 😛 so if anybody has some similar issue they can try that too (remember to back it up!!).