I outgrew and stopped using DVDs for storage a while ago. They are a real hassle to burn and store.
Big hard drives have come down dramatically in price, so there's little reason not to go with them instead of DVDs.
The first thing I'd look into is either replacing smaller drive(s) with bigger, and/or adding a second (or third or whatever) hard drive to your computer, assuming it's a desktop. Most have room for at least one additional drive. Some have room for many more.
My primary desktop has a 1TB drive for softwares, general purpose and a partition for Photoshop to use as a scratch disk. There's a second 1TB data drive used strictly for recent image storage.
If yours is a fairly recent computer, with SATA connectivity, it's very simple to install a drive and get it set up. And you can get 1 or 2TB drives pretty inexpensively. I recommend using "enterprise class" hard drives, in particular. They are a little more expensive, but typically have five year warranties and are designed with large, server applications in mind, where they run 24/7/365. They consume less power and are better built to give long, long service life. Both Western Digital and Seagate offer these types of drives. I'm sure some others do, too.
On location I use a laptop with a 750GB external drive, which attaches with USB or eSATA. It's convenient, but not quite as fast to write to and retrieve from when using USB connectivity, compared to an internal drive.
Now for backup current work and archiving previous years' images, I currently use three Netgear NAS servers
on a simple, wired network. Each contains four 1TB drives, but using RAID X they only provide for 2.7TB storage space per server. But it's worth it since each server also backs itself up. I can replace any individual disk in the server if it crashes, and the server will automatically rebuild the new disk without any data loss.
I'm going to be adding a hot swap tray to my desk top, so that I can use 1 and 2TB drives for incremental backups to be stored off site. An off site, secondary backup might be important if I ever have a disaster wipe out stuff on site, such as a fire or flood or theft.
Chances are the "corrupted" files are fine, but you just have way too much on the hard disk. Once you get about 90% full, things slow down and errors start to happen. Move stuff off the disk and see if that helps. Hopefully the images are fine. You might need to run an image rescue/repair software on them... such as Lexar and Sandisk sometimes provide.
Another common cause of RAW file corruption is when transferring them. I had issues with a memory card reader causing occasional corrupted files when the reader was attached to a USB hub. It works flawlessly when connected directly to the computer, though.
I've gotten more ruthless about editing, as camera resolution has increased and image file sizes have grown and grown. Currently using a pair of 7D and 5DII, so I fill up drives and NAS servers pretty fast!