How to shred carrots without them getting mushy
The best method I've found for this is to use a vegetable peeler. This results in nice, beautiful ribbons with no mush at all. 

I recently used this method when making homemade cole slaw and it gave a wonderful texture and presentation. I'm not sure if it's the size you're looking for, but given how well it worked with the cole slaw I made, I can't imagine it couldn't be used successfully for a salad. 

To do this, you'd simply peel the carrot, discard any dirty peels, and then continue to "peel" the carrot. The strips you get will be even, clean, and without any mush at all. 

If this method is not to your liking, certain food processors have attachments which may be able to give you results similar to the shredded carrot you can find pre-packaged in the store. Yet, I couldn't say if this would result in a mushy consistency, as I haven't used that method myself.

Update: Note my comment below. There is a device known as a "carrot shredder" which may be what you're looking for. It looks like a vegetable peeler, but has notches on it. This results in shreds instead of ribbons.