I think what youre looking for are closed back headphones that will prevent noise from leaking out? Noise cancelling will stop sound coming into your ears from the outside, although that could work in that you wouldnt be able to hear your girlfriend shouting at you.
Personally I'd recommend getting a set of over ear closed back headphones with an external mic. I have Beyerdynamic DT770 Pro 80 Ohms and theyre still pretty popular afaik:
With this attached.
ModMic USB 2 takes everything great about the original design and delivers it all in a single microphone capsule. The USB 2 features professional grade mic quality with a noise rejecting cardioid pickup pattern, making it the most advanced wired ModMic we’ve ever made. The cherry on top: We’ve...
antlionaudio.com
Had them for years, my partner is very sensitive to noise and has never complained when Im wearing them.
Whats your budget? Maybe either
@Colif or
@anort3 may have some suggestions. I've never used a gaming headset so cant help there unfortunately.