Here's an update. I've caught Latios and Rayquaza! And I didn't use my Master Ball!!!
Saturday, I finally got to use my EON ticket. I went to the Southern Island expecting Latios to be easy. Boy, was I wrong! I didn't catch him until Sunday sometime! One time, he actually had me use up every Ultra and Timer Ball I had, along with a few Great Balls out of desperation (just to tease me, he let a Great Ball nearly catch him...). That time, Gardnevoir actually used up all of its PP in Hypnosis and still no capture. The next time, I went and bought 20 more Ultra and Timer Balls...and caught him with like the 4th Ultra I tossed. Stupid Legendary Dragon....
Last night, I went ahead and went after Rayquaza. After spending a good half hour or so trying to get over those cracks (stupid Mach Bike...) I finally got to Rayquaza. I guess the difficulty in getting to him made me lose my head, but I was actually trying to paralyze him at first! Or rather, paralyzing him and trying to catch him. After 3 times of doing that, I ended up bring up my Gardnevoir. Then, Rayquaza used Rest and it dawned on me that putting him asleep would be so much better.
I accidentally killed him again that time. The next time, I started with Gardnevoir. Ray used Outrage. Gardnevoir survived Outrage before, so I wasn't worried...until I saw that it was a Crit....
The next time, though was the charm. Caught him right quick.