The A350 is a fine camera. I'm sorry to read that you're not happy with it.
Sony has made advancements in sensor technology, and it's newest Alphas have lower noise and greater dynamic range, but it's also gone off in a slightly different direction. Instead of reflex cameras, Sony now only makes cameras that look like reflex cameras, but instead of an optical viewfinder, it uses an electronic viewfinder. This helps you in one significant way, and that is that the mirror doesn't move, thereby reducing the noise the camera makes when you press the sutter button. Whrn you press the shutter button on your A350, several things happen: the mirror flips up out of the way, the two curtains of the focal plane shutter expose and then cover the image sensor, and the mirror returns to block the image sensor and reflect light up to the viewfinder. So since the mirror in Sony's new cameras doesn't move, a major source of the sound you object to, is gone.
Sony's current product line inclused the A57
, which use a 16MP image sensor, and the A77
, which use a 24MP sensor. These will all outperform your current A350 in terms of noise and image quality. The A37 uses a smaller battery than the one(s) you already have, but the others can use the batteries you've got now.
Sony doesn't currently have a 'Full Frame' dSLR. It's A900 and A850 have been discontinued (though they may still be available at some retailers), but the rumored replacement is rumored to be available in the comming months.
You seem to have some interest in a 'Full Frame' camera. For the most part, a 'Full Frame' camera body only increases the potential for shallow Depths of Field, but is bigger, heavise, and more expensive. Can you go into more detail about what you expect to be able to do with your next camera?