When my work visa is up, I was told that I could get a tourist visa without having to leave Kunming. Is that true? If so, where do I go to do this?

Not 100% sure about the rules now, but in the past you could have your employer apply to the Visa PSB to change it to a F visa for the time you needed to get your things in order before you leave.

30 days 'L' visa is normaly done at the PSB.
Be careful, this is 30days after the date of application, if you apply too early before your work visa expires, you lose time here. However, you must apply before your work visa expires.

Last time I did this, my employer accompanied me, this may have been a requirement. That was two years ago.