Well, some stories can work that way (the seriousness or the beginning with diapers), although you must be wary because a LOT of plots have been done this way. Some were good, some...not so good.
Might I suggest you try a full-fledged character profile? Similar to RPing, but really get into this character. What does he/she eat? Left-handed or right-handed? Little, small details like this will bring any character to life if done correctly. And most importantly, why is your character wearing diapers? Simply saying "because he/she likes to or has wanted to since they were a kid" won't cut it. Maybe throw in an experience they've had in the past or something of the like.
I can't tell you which plots/characters are perfect for diaper-related stories, mainly because nothing is easy or perfect. One thing I CAN suggest, however, is that you take a pen and paper (yes, pen and paper. NOT your keyboard with a blank Word document) and write for fifteen minutes straight. If you mess up a word, cross it out and keep writing nonstop. Anything and everything that comes to your mind, write down. Even if it's something along the lines of,
"I want to write a story involving diapers, but it's too damned hard! What am I going to write about? I could writer about...." and just keep building off of it. Sometimes this has been the key for me finding a unique story to write. As for the plots, you should definitely do some research. Go to other forums and see what's been used and what's getting good feedback.
I really hope this helps you out!
If you need more help, I'll post it here or you can just send me a post or message.
