First, I think your coach is wrong: [weightlifting, properly done, is excellent for boxing][1].

Second, I think your assumption that your legs are strong is wrong: running is not a particularly good way to develop leg strength compared to methods like barbell squats and deadlifts.

Third, I think punching with weights in the hands or on the wrists is a bad idea for several reasons. First, the resistance is in the wrong direction (mostly downward, in addition to against the direction of the punch). Second, it raises the risk of elbow injury if you extend the punch fully. Third, because the activity of punching with resistance is very similar to punching normally, it can interfere with developing speed, speed-strength, and technique efficiency in general. Fourth, why are you punching with dumbbells when there are other ways to get stronger and more powerful in that movement?

If you want to punch harder then you need to develop basic general strength first. This means you should start [a basic strength program][2], probably with a barbell and weights, using [fundamental exercises][3] like deep squats, deadlifts, pull-ups, and dips or presses (bench and overhead). Once you reach some basic strength benchmarks (deadlifting 1.5x bodyweight, easily squatting many times with a barbell weighing as much as you do), incorporate power versions of the Olympic lifts like the push press and power clean. 

A general strength program will be enormously more productive for you than punching with dumbbells or wrist weights. Try it.


  [1]: http://martialarts.stackexchange.com/a/855/347
  [2]: http://fitness.stackexchange.com/questions/12405/adding-strength-training-to-yoga/12406#12406
  [3]: http://www.t-nation.com/training/11-myths-of-warrior-training