Yes, it is normal. When you go to Craigslist, you can search for the item/s you want. Then it gives results - from ANY craigslist listing that matches. People shopping your local craigslist is not automatically a scam.
Second, money orders can easily be counterfeited. I would not accept a money order if I were you. Paypal is very secure. Just make sure you specify "funded" amount only (meaning money in the account or linked to their bank account - with credit card, they can do a "chargeback"), then ship through paypal - more security when you ship to their listed address. Plus, delivery confirmation is very inexpensive through paypal shipping.
If you sell to someone not local, then yes, they should pay the shipping costs. Another poster said use a flat rate box; that is usually MUCH more expensive than weighing the diapers and then putting them in a proper mailer. gPants are pretty light, and might not cost more than $4 or $5 to ship.
gDiapers (and many cloth diapers) are expensive items, and resell at more than 50% of retail, if they're in good shape. So if you priced yours less than 50%, I'm sure you got LOTS of responses! The person that offered more might be a scam, though.
Use paypal to accept "funded" payments, ship through paypal with delivery confirmation (and possibly even Signature Confirmation), have them pay shipping and don't ship until you get all of that. Most importantly, paypal will NOT protect you if you use any address other than the one listed on their paypal account. if they ask you to ship elsewhere, move on. Or just deal locally.
But it IS very normal for people in other parts of the US to offer to buy your CL items, especially for something like diapers.
BTW, why does a woman of three older daughters have gPants for sale?