If there's a warranty, the cost of shipping may take care of the tedious work for you.
Otherwise, yeah, pick up a USB to SATA adapter, which is an external drive without the drive. They're usually about twenty bucks and nice to have around regardless, for those days you remember that old computer you haven't seen in years and want to know what's on the drive.
Anyway, scour the existing case for stickers that hide screws and remove them. Yank all the screws and take the case apart. Remove the drive and plug it into the adapter, and you should be back running.
I'd recommend copying everything off immediately, just in case there's a problem with the drive itself. It's rare, but if there was a "head crash" (the sensor that reads the data scratches up the disc inside), the damage will just get worse over time, kind of like a pothole in the road.
I don't know how reliable my information is, but I've been told by a few people that the hardware in the external drives is usually lower quality than if you buy a drive without any enclosure. That's another handy use for the adapters.