The key is that you can use ~/.ssh/config to set up an alias of github, such as github-my-org (using your new account and new key to connect) and then you will need to replace the git remote string:
from git@github.com:username/repo.git
to: git@github-my-org:username/repo.git
So that a particular repo will use your new key to push to Github.
If ssh keeps trying different private keys than the one you specified in ~/.ssh/config, then you may need:
IdentitiesOnly yes in the config to only try the one key file you mentioned.