You will have to call Nintendo and tell them your system is broken and they should help you by transferring your old account over to the new system (need to know your old system's serial number and new one) and you'll have all of your games again just without the save data. I have heard others say they got the run around from Nintendo but most people have had good experiences. I personally trust that Nintendo will help you if your old system was linked to a club nintendo account since that helps prove ownership and whatnot. I also have a scan of my receipt with the serial number just in case I have to send them that.
Women just want to take your money and eat all your raisin bread. Parking meters! And you're walking around! Ha ha ha That's so funny.
Register your 3DS on clubnintendo.com and link your Club Nintendo account with your eShop account.
Your 3DS won't have any nightmare scenarios in all likelihood.
But should you have issues, call the Nintendo Customer Service 800 number. They keep a record of your eShop purchases, as well as your system serial number, provided you have completed those initial steps.