if my screen is replaced, my touch id will continue working?

It will if Apple replaces it, and even if they broke it they can fix it or they would replace the phone. All bets are off if anyone else changes it. A competent technician can replace a screen without breaking Touch ID, however, there is no certification requirement for independent technicians so it's a crapshoot.

There's no guarantee if you let a 3rd party do the repair.

If you did then Apple will not service that iPhone.