I used to have that problem...only able to swim in shallow water. But...it's actually just a matter of take the "courage" to swim in the deep area. As long as u r along the side (so if anything you can hold the side), there is no difference in swimming in deep/shallow water. Actually the body floats better on the deep water.
Wave pool is different story....I wouldn't dare to try the wave pool if I can't swim ....
You would be fine in the deep end if you can swim in the shallow. Just hang on to the side.
I'm not good at swimming, I'm able to swim in deep water coz I know I can always swim to "hold on to the side" if I run out of breath. But in ocean...I am still afraid to try yet..