It does have a seperate tuner (The slingbox pro HD that is). However you must wire the installation correctly in order to use it. Right now the reason your TV changes at the same time you change your slingbox is because your on the input you connected to the single cablebox you have. Thus the cable box can only be on one station at a time.
In order to have an independant sling experience you also need to split the incomming cable tv line with a 2 way splitter. One output of the splitter goes to the existing cable box as it is now; while the other output connects to the slingbox pro hd's coaxial input.
When you place the HD pro's input on "coaxial" you get ATSC or Qam or analog signals from your cable company independant of the locally connected tv.