Moved to new domain. Should I delete old database and files?

Hi everybody,

We recently moved domains due to a rebranding. The process we followed was this:

1. Register new domain.
2. Add domain to our hosting service.
3. Copy files in FTP from old domain to new domain.
4. Logged into new copy of wordpress and updated old domain to new domain name throughout files.
5. Set 301 redirects from old doamin to new domain.
6. Found out that we also have to set up new database. When we copied files we didn’t deploy new database and instead used old one which was a mistake. 301’s in .htaccess file set up a circular reference which caused some issues.
7. Set up new database and this got rid of circular reference.

So now I’m wondering if we need to delete old database and wordpress version at old domain. If you visit our old site https://www.constructionei.com it automatically pushes to new site https://www.buildinghunter.com. However do I now need to delete old database and version of wordpress so we don’t get in trouble for duplicate content?

Google Webmaster tools recommends we hold onto the old domain for at least 6 months, but I’m not clear if I should take down all wordpress items from this domain and just allow 301 in place. Anyone gone through this before? Any ideas?