All three fit and would make sense, however the most commonly used is likely "die out."
"To make matters worse" fits in a written setting, "What is worse" certainly does not. However, in spoken language, something along the lines of "What's worse is that every 60 years or so..." would be acceptable.
In your first sentence I'd prefer the word order 'die off naturally' if you are referring to an activity that is a natural process.
In my opinion, it would be more natural to use 'die out' in reference to an animal or creature, rather than a plant. 'Die away' would be more appropriate when referring to the lessening volume of sound, for example. It sounds odd
I agree with Yankee... "Die off (naturally)" would work better in connection with a plant. I stand corrected.
However, I assume this might be a dialect or regional issue. The sound reference seems funny to me (dies off seems like the appropriate term in that situation).
Angliholic is quite find of asking some difficult questions on usage. I really wasn't sure quite what to make of 'die away', but it didn't sound as natural to me as 'die off' in the context. I had also thought of something such as light for 'die away'.