Calculation of an autocorrelation function
I guess you've done almost everything right, but have a problem at the calculation of the expectation value regarding $t$. You should calculate the expectation value of the cosine function. Sadly, it does not simply "go away" as you wrote.

Have a look at the Wikipedia page. There you can find another, more explicit, formula for the auto-correlation function of a function $f(t)$:

$R_{\textrm{ff}}(\tau) = \int\limits_{-\infty}^\infty f(t+\tau)\bar{f}(t) \, \textrm{d}t$ .

(Note, that compared to the Wikipedia page, I have taken the liberty to use the variable $t$ in the integration instead of $u$, which would be the mathematically more accurate version.)

As you can see from this equation, you "integrate away" the dependency on t, and indeed you should be left with a function that is independent of $t$.

Note that there is also an version that does not go to infinite times, but is constrained to a period $T$. Maybe this version is more appropriate in your case. However, the same holds for this version: $t$ is integrated away and should not be a variable in the resulting formula.