I am an American who applied and received an L visa in Los Angeles, but my visa will expire soon!?!

You can't renew your L visa in Busan before the visa expires, if you don't have a Korean Alien Registration Card with at least 6 monthes remaining validity, you can't get a new L visa in Busan, either.

What if I make a trip over to Japan, will the Chinese Embassy work differently there??? I am really trying to find a solution, so that I don't lose out on the $140 US Dollars this Chinese visa has already cost me!?!

Even though you go to Japan, you can't renew your China visa, once the visa expired, you have to apply for a new China visa in Chinese embassy in Tokyo and pay for the fee again. Now if you don't use the present China visa to enter China, the 140 US$ will lose. You can't get a new China visa in South Korea because you don't have ARC with at least 6 months remaining validity.