"I feel that my body is not comfortable."
They should just change this to "I feel uncomfortable" and then modify the correct Mandarin. Like those before me, I agree that this English sentence is weird-sounding.
Grammatically you can, but you shouldn't. Adding a 了 here would be perceived as a change of state for the verb. 我病了 is a change of state from not being sick to being sick; 我觉得XX了 would be a change of state from not feeling something to feeling it.
Chinese is quite flexible that we can also say 我冷了 (I am cold) to mean I feel cold. By this we are indicating that I don't feel cold previously, but now I feel it.