Do you have insurance? If so, call them and ask. If not, do you have a general doctor? I didn't have insurance for the longest time. My long time family doctor gave me free samples of the cymbalta. The company actually had a low income program that would have helped me if I had stayed on the med (Hubby went into the AF and I got insurance so I didn't need to use the program).
Zoloft is a commonly prescribed drug so I'm sure it'd be easy to get samples. And I'm sure Walmart has the generic form for cheap. Going to the doctor without insurance can be expensive but call around and see if there is a place that does a discount for cash. My Dr without insurance only charged something like $40.
Call and ask. I had it for pregnancy but couldn't be on meds while pregnant so I don't know about psychologists, but I'm sure it will cover it.
If not, there are usually low income places that will do like a sliding scale payment based on income.