I work as an assistant to a guy who owns 8 apartment buildings/25 apartments, doing, in part, lease and contract reviews. I don't see at all why you can't start moving into your new place. Assuming that the notice you gave to not renew your current lease was before the notice due date (generally 60 days before the lease expires, check your lease's language, as there could be a penalty to not giving notice before a certain date -- forfeiting your deposit, etc.) and that you've worked out the move-in date with your new landlord. And as for the place you're moving out of: even if your furniture and belongings aren't in it, the place is yours for the duration of the lease. But I'd probably notify the relevant people of your imminent change of address.
All they need is a 20-day notice, I've effectively given them a 30-day notice. Lease renewal would occur on 5/1, if I were to have renewed it. Move-in date has already been worked out with the new-landlord (who, so far, seems much more pleasant). It doesn't look like there are any issues?