Following the repair guide referenced below, you may find the problem is simply a ribbon cable connector which needs to be reseated.
If not, search on eBay for another camera of the same make and model but with a different fault. You should be able to get one quite cheaply. Then learn how to disassemble it using the guides here and any other guides or videos you can find on the Internet. You should then be able to swap out the shutter and lens controls between your camera and the one from eBay. From a quick look at the guides it seems you may need to swap the entire front cover of the camera with the shutter and zoom controls attached, but that’s not a problem.
Having fixed your camera you can then try a more radical disassembly of the dead shutter and zoom controls to see whether they are fixable - it won’t matter if you break them further in the process. You might even be able to fix the camera from eBay and sell it for a small profit!
[[Topic:Sony Cyber-shot DSC-WX350]]