Sounds like you're relying on a high brace, which is not ideal for a lot of reasons. For starters, it puts way more stress on your shoulders. It also tends to put more of your body in the water, which makes it more likely you'll be flipped by a rogue wave as you try to right yourself. Lastly, it can be harder to right yourself using a high brace because you're already most of the way over.
Instead try to focus on using a low brace (using the front face of your paddle). Low braces are nice because you can easily turn them into quick forward strokes or draws. They also tend to keep you centered over your boat more.
Another thing to keep in mind is that the paddle is just there for balance. If you're relying on the paddle to right yourself, you're doing it wrong. In addition to using a low brace, focus on snapping your hips to get back upright. Otherwise you'll just lay on your paddle as you get drug over rocks.
An easier boat will probably make you less likely to flip over in the first place, but you have to perfect your strokes if you want to progress in this sport. Take some time out and get dialed on flat water.
"A river is more than an amenity, it is a treasure."
-Oliver Wendell Holmes writing for the majority in New Jersey v. New York