It's not the end-all solution but
303 Fabric Guard will make the awing absorb less water so, when there's a little water, it stands a better chance of running off. In addition, it has fabric protecting qualities. By the way, there is also a fabric cleaner from 303.
Unfortunately, once your awning has already created a belly, you'll need to replace it (or use one of the methods in the above posts) to get much hope of using the Fabric Guard to make water run off.
Note:
The above link is just the first one I grabbed from Google. You might try Amazon, instead.