Use noise-attenuating headphones rather than noise-cancelling headphones. A pair of properly fitted IEMs or a good pair of closed headphones will reduce background noise by ~25dB simply by providing a physical barrier to noise.
The distracting effects of music largely disappear if you listen to repetitive instrumental music - house or ambient music is ideal for this.
If you're concerned about noise-related hearing loss, Canford produce a range of high-quality headphones with integrated limiters. The limiting system was designed by the BBC, to enforce safe noise exposure for employees who use headphones all day.
While I agree that open plan offices are generally a bad idea, I think that OP is rather uninformed about the use of headphones.
To build on your second point, the key is more "instrumental" than "house or ambient", though combining both is indeed ideal. Soma.fm's "Groove Salad" online radio station is a good example of such music.
The distracting effects of music largely disappear if you listen to repetitive instrumental music - house or ambient music is ideal for this.
If you're concerned about noise-related hearing loss, Canford produce a range of high-quality headphones with integrated limiters. The limiting system was designed by the BBC, to enforce safe noise exposure for employees who use headphones all day.
While I agree that open plan offices are generally a bad idea, I think that OP is rather uninformed about the use of headphones.