You will be very much happy with a Xonar Essence STX and a DT990, even the 600 ohm. I was there for quite some time, and yes, the STX will drive even 600 ohm headphones, but the problem is, how well. Since a year ago I was in your exact place (I used grado sr-60's through onboard sound, then went to the STX with DT880/600's), I can tell you first hand you will think they are the best damn thing you've ever heard in your entire life, and you will be really happy. Then after time passes and perhaps you have a little extra money, you will accidently hear something better, and want to upgrade.
My best advice is to upgrade slowly and enjoy where you are at with each step, and be fine with being a little naive. It gets unhealthy for one's wallet and mind reading this forum too much, especially at first
My advice for you - buy the STX and those cans, get the 600 ohm, you will love it.
Now I see you are also curious as to why someone would want or need another amp, even though the stx has one built in? Yes, the stx will drive the dt990/600's, but there is a great sound improvement when going to a dedicated amp. To be more precise, there is a great improvement when you switch to a QUALITY amp. You can read countless threads here like I did, and from what I took from them, the amp in the Xonar Essence STX is unmatched
for its price point. Some even go further to say you'd need to spend at least a couple hundred on a headphone amp alone just to outdo the STX. The STX is a great buy at 200 bucks, you get a solid DAC and a solid Amp - its the perfect solution in my opinion for someone new to this whole thing.
However, consider that the amp in the stx is a simple (albeit high quality) chip the size of your fingernail, and you can see how that may struggle to perform as well as say, a 500 dollar, hand made woo audio 3 tube amplifier. There are many threads as to why this is true, speaking in terms of voltage delivered, etc. I'll let you search those on your own if you wish.