I've two separate domain
Well, humor me. Let's try setting a cookie prefix for at least the new installation, please! Don't forget to clear your cookies in ALL browsers you use.
The other items that could relate are any settings for session timeouts - php for example. Go directly to using URL:
Look for session information.
Am getting close to running out of suggestions without knowing hosting provider and type of package purchased! :|
'spirit of sharing', Ken