35 minutes is the legal connection time. Remember that doors shut approximately 10 minutes early if possible.
Delta will not automatically protect you...you could be on standby hell. But, you are one of the lucky ones with the new policy. Delta will confirm you on the next flight for free if seats are available.
If your flight comes into E and you are connecting to ASA, you should be able to make it if you hurry. I would definitely hurry.
marc