Yes, what you want to do is create a new primary account with its own gamertag for the 2nd PS3 so that when it boots up it will load that profile, this will also let people play online games at the same time on both PS3s. Then create a 2nd profile on the new PS3 and use that second profile to log into your original, old account and download games to the new console from your download list. Then you can take the password off the second profile (so they can't buy games, rent movies, etc.) and as long as that 2nd profile is on the new PS3, you can play any of those games while booted up in the new, primary profile.
Prior to a certain date, you could gameshare a game on up to 5 separate systems, but after a certain date you can only gameshare between 2 systems. As long as you have not owned a previous PS3, you should be fine. One thing which cannot be shared is video rights, any movies you rent or purchase can only be watched on one system.