Functional inexpensive yixing-like pots do exist, I have a very small one from ebay which cost $10 that I enjoy, and I know several other members bought them and like them too.
But, I just checked the seller I bought it from and they don't have them anymore or any that really look similar.
There's some chance you could get a pot which alters your tea in a bad way like robbing it of the high flavour and aroma notes (probably the greatest risk) or that smells bad...
If you want an upgrade to what you're using now made of clay try getting a gaiwan or spouted "easy gaiwan" like one of the ones here:
http://yunnansourcing.com/en/8014745-gaiwans
They're inexpensive and a safer bet than going for yixing in your price range. Most are smaller than you're asking for, but most people when they start drinking better tea get used to smaller cup sizes too
