we recommend using M mode for badminton.
in an indoor gym, the lighting is pretty constant near the ground. experiment and find a ISO/shutter/aperture that gives a good exposure and stick with it.
using M has the advantage that the shutter speed and aperture will be predictable. if there are sudden bright spots like if the shot happen to see a light, it won't make the subject totally dark. where if you use Av/Tv mode, it will close up the exposure and the subject will be underexposed.
in most local gyms, you should expect to use f/2, ISO3200, and depending on the gym itself, shutter speed from 1/250 in dark gyms, to 1/640 in brighter ones.
in international stadiums, there is an extra stop or two of headroom.
badminton shooting is fun but very difficult as the action is super fast, and the gyms are usually super dark. it stresses both equipment as well as the photographer's skills.
our recommendations for lenses for badminton in local gyms are always primes. primes that will do f/2. the best that we have found are: 50mm f/1.4, 85mm f/1.8, and 135mm f/2L. you cannot go wrong with any of them.
keep trying!