Need new headphone setup.

I would just go over to Drop (that's what I did).  There you can pick up a decent DAC, headphone amp, and a set a cans for under $1k. You can of course spend more too but you don't need to for some decent stuff. 

1+ Soix

Agree with the Audeze recommendation.  I enjoy my Audeze LCD-X.  Easily run off an iPad or phone.  Being over-the-ear, they’ll be a real change from in-the-ear monitors.  They are heavy but, unlike other ‘phones that I’ve had, not fatiguing.  Stellar review in The Absolute Sound by Arnie Nudell (the genius behind Infinity and Genesis); it’s an older review (Sept. 23, 2014) and they’ve since been updated with fazor technology. List price is $1,190, but Music Direct currently has an “open box” set available at $999.  Best of luck with your search!

Good luck with your HD600 and Cobalt!

I was gonna say that the Cobalt is great. I use a Black myself but has listened to the Cobalt in a store and it is better.

If you ever want better headphones I suggest trying Focal. I think the Elex from Drop is available for $700, cheaper if you buy used. I have Beyerdynamics Amiron Home which were said to be slightly better than HD600. There is also DT1990.

You can also look around in the headphone forums for more ideas once you get hooked. For example in Reddit. 

I'm sorry, but all "ear speakers" absolutely do not sound fine if they are used with "top shelf" components feeding them. If that were the case, then the same would be true for typical home speakers and that's clearly not accurate at all. As for the OP, you do not need a headphone preamp at all. You can use some preamps as headphone amps if they have a jack, but you don't need a preamp for a headphone amp, if that's what your asking.

Can they be open headphones which leak sound or do they need to be fully closed?

If you can use an open set then consider the HiFiman Ananda with the Topping DX3 Pro+ amp/DAC/Bluetooth combo.

For a closed headphone, there are not a lot of great choices.