I have done a few multileg flights, but the ACARS don`t reset when I select a new route and so on..

Why is that? Is it something I am doing wrong, or is it an easy fix to reset the ACARS?

As outlined in the Introductory section of the manual on Acars, there is a 900 second "Reset" circuit that runs after shutdown.  If you leave the aircraft at the gate for 15 minutes while prepping, fueling and loading and maybe getting a snack and some coffee for you next leg it should reset once the 900 second counter expires. If you save the flight right after shutdown, the timer state is also saved so you will have to wait for the remainder of the clock to run out before it resets for the next flight.