How can I use this lightweight piece of furniture?

Not sure of the name but it is something akin to “sailors’ putty”. It’s used to keep things in place while underway at sea. Check with a store like West Marine.

Poster putty available at Walmart will work. It looks like bubble gum. I roll it out very thin and apply along the edges of the item to hold it in place.

Not sure I posed this question correctly. What am I supposed to put putty on?? Thinking about cutting the shelf in half so it has more stability!

Sounds as if you could use something under the front two legs to raise them a bit so that the shelves (the whole unit actually) ends up a bit higher than the back. I've done this with those little self-adhesive pads for under furniture feet and it worked well for me. On larger furniture pieces I sometimes use a plastic shim to achieve the same results. Or you might try placing a heavy piece of decorative pottery (fill with cement if it isn't heavy enough and you don't cherish it). All the best.

If you have w to w carpet your room is likely bounded by tac less which could be lifting up the back of the unit. Try using some furniture glides under the front legs to level your shelving unit. After you have your unit level you can use museum putty to hold delicate items to the shelves.

V Smith is correct about the carpet strip raising the back up. So something under the front feet should level it for use on the carpeted area if you don't have an uncarpeted area where it will work for you. (Try it on an uncarpeted area to see if the legs are really uneven. My bet is they aren't.) You can also move the unit a tiny bit out from the wall to try to level it because those tac strips are right at the edge. (It looks like it's pressed up against the baseboards.) If it's an antique and not merely "old," I wouldn't cut it up. If you don't like it, sell it or give it to a family member and get something more to your liking!

Thank you to everyone for the suggestions!

Here's someone who cut theirs in half and painted it. It looks great! https://ourhousenowahome.com/2014/02/corner-shelf-redo.html