First of all, you need to check and see what fabric has been used to make the running clothes that you are running in. If your tags all say cotton, the very first thing you need to do is hit the mall and grab some new threads! In all seriousness, running in cotton can be a bit of a nightmare for many reasons. The first is precisely the point you are making - cotton retains water and stink in a bad way. Furthermore, while some cotton tops might seem airy and lightweight and breathable, the reality is that they hold water. This can make for a horribly uncomfortable run, especially during the summer or when running in hotter conditions that promote a lot of sweat and water loss. Cotton clothes will hold on to water, which results in added weight that you will be carrying around, which can ultimately lead to slower running times and just a general feeling that you are carrying a weighted vest on your back.
If you are already wearing clothes made out of technical fabrics that are designed to wick sweat, help keep you dry, and stay light and breezy, then that is a step in the right direction. If you scour the stores and look online, you can also find running clothes that are specifically engineered to help fight stink and odor. Odor-fighting running clothes are designed using a small amount of metal in the fibers - not enough to weigh you down, but enough to help combat stench. However, if you tend to get sweaty, then your skin is creating the bacteria that is responsible for the stench you are smelling. This bacteria can easily get trapped in the fibers of your running tops and shorts (especially if they are not the specialty odor-fighting clothes). In order to really get the stink out, you are going to have to do some overhaul. Try first pre-soaking your clothes in a serious solution of water and baking soda. If you are familiar with the trick of keeping baking soda in your fridge to help soak up the stink, then the same principle can be applied here to your laundry.
The best way to fight odor is to continue doing exactly as you are doing and washing your running clothes IMMEDIATELY. The longer the odor stays in your clothes without being washed, the harder it will be to get out. Try using a specialty sports laundry detergent (specifically designed to target body odor oils and bacteria) and an in-wash odor eliminator.