We just got one for Sheba, as she's a "shoveler" and tends to make quite a mess around her litterbox. I was worried she wouldn't adjust to the litter box, so we put the new one out w/out the lid first (it's much higher than the old one) and then once she got used to that, put the lid on. She took to it no problem. Now I just have to get a new mat, as the one we have, she now clears when jumping out of the box and tracks litter through the bathroom!
So if you're worried, maybe try "transitioning" from regular to covered, then it won't be a complete shock to your current 3.
Oh, and we bought ours from walmart... about $13, and very big.