Evaluating the autocorrelation function of a telegraph signal to get exponential decay?
A semi-random telegraph signal is one for which $X(0) = 0$ and $X(t)$ equals $0$ or $1$
according as there have been an even number or an odd number of arrivals in a Poisson Process in the interval $(0,T]$.  A simple physical model would be the least significant
bit (LSB) of a digital counter that is counting the Poisson arrivals.  A random telegraph signal is almost the same except that $X(0)$ is equally likely to have value $0$ or $1$.
(Feel free to translate this to signals having values $\pm 1$ if that is more convenient
or more in accordance with your notation).

With this as background, you are encountering problems because your simulation is
not modeling the random telegraph signal correctly. You should instead simulate a 
Poisson process and obtain the random telegraph signal from that.