Are you near a Guitar Center or Bestbuy ( or another electronics retailer with a good headphone selection)? Two popular options would be the Sennheiser HD 558s and Beyerdynamic DT 770 Pros. The bestbuy near me had a Magnolia section which featured a few mid to high end models from Sennheiser including the 558s. The Guitar Center near me also had a decent headphone selection and it happened that a DT 770 Pro 80hm was on display.
Both are very different headphones. The HD 558s lean to more of a balanced sound signature and they are open back. Open headphones tend to have a wider sound stage at the cost of bass and sound isolation. Since they are open they have better insulation and airflow for , letting your ears breathe and they won't get as hot and sweaty as closed backs can in a short amount of time.
The Beyerdynamic DT 770 PRO is a more expensive model and comes in the 32, 80, and 250 ohm versions. The lower the impedance the easier it will be for your source to drive it. 80ohms is known as sweet spot for this pair of headphones and it should be able to be driven to decent volumes with your onboard amp/dac. If you wanted more juice out if it you could always look into an amp later on. This headphone is definitely for your bass heads, and the "fun" rating can be described as phenomenal :lol: . However the great bass extension comes with a cost to the mids and highs. Despite being closed back the 770 PROs have a decent sound stage and can be quite immersive when used for gaming. Granted if you're playing specifically to listen for positional queues such as footsteps, these might not be the best choice due to the slightly over powered bass. Comfort wise these are also great and have velour pads like the Senns.
If you went with the Beyerdnamics it would probably push you over the $150 budget. You could get a cheapo 3.5mm mic and 3.5mm male to female extension cable to use with the 770 Pros (attach with cable ties). It would still work fine, but wouldn't be as high quality as the ModMic.