Define the "partial utility function" as how utility changes with x holding c constant (i.e. U(x) at a particular value of c). Changes in values change this partial utility function, but they never change the full utility function U(c,x). A real-world example: if you prefer to vote for the candidate that gets the most votes, then your vote will depend strongly on the other voters' values, but this preference can still be represented by a single, unchanging utility function.

I don't understand your second paragraph - why would having values as a variable be bad? It's certainly possible to change the utility function, but AlexMennen's point was the future values could still be taken into account even with a static utility function. If the utility function is constant and also depends on current values, then it needs to values into account as an argument (i.e. a variable).