The process is certainly a pain. Some places won't give you applications until later in the year, like May/June, or even as late as August. I told them it took 8 weeks for applications to get approved on my end at my school before they would even send them back to the programs. While a bit of a white lie, it seemed to work and got the ball rolling sooner; every place I told this to sent me an application.
Regarding getting a spot, I didn't get the impression it was very competative, even at bigger name places because they can take so many students. However my impression is probably biased because I got everything in early and didn't get turned down anywhere. I would recommend sending in your apps a couple of weeks before their earliest date, unless of course they say they'll throw them out if they get it early. Some will approve you ahead of time, others will just hang onto it until the first date, in which case you'll be the first application they have, and thus you'll get approved. A good rule of thumb- there is very little in life you get penalized for being ahead of time.
There is no one source for away applications, each department has there own. What you're doing is the best thing possible to get the apps.
