Congratulations on the scholarship! Your parents must be both proud
and grateful for your hard work to get that.
In general, yes--for music, discrete DAC's are the way to go versus a "codec" all-in-one chip.
If you can go to $180, you have a lot of options....and maybe an even tougher decision as to how to balance your music vs. gaming requirements.
Not being a gamer (my son is a sophomore in college, so I am a bit past the median gamer age, I guess
) I have no idea what capabilities/compatibilities you need for the games you like.
From what I have read, I would take a hard look at the Sondigo Inferno card, available for $80 at newegg (uses the AK4396 DAC chip, AK's current flagship.) It has an ASIO driver available (good method for music via Foobar) and supports EAX™1.0&2.0, A3D™ 1.0, and DirectSound, according to:
http://www.sondigo.com/inferno
Next, find a used headamp here on Head-Fi. For $100 or so, there will be lots of choices available. If you want a strictly desktop solution, you should be able to find a "desktop PIMETA" with a wallwart power supply around that price, or for maybe a bit less, you can find various GoVibes with an Elpac for use on AC power back in your room and also have battery power for portability. All depends on what is available at the time you need to buy.
Closed cans will be a "matter of taste" for you. I am a Senn HD600 fan, but have a pair of HD280's that I like to use on the road with my notebook--I can drive them directly with my notebook's sound card (which cannot drive the HD600's very well) and block out any other noise. It may not be so easy to find a closed model that you like.