When I use TLT and roll 'X' hits, I know I cancel all dice results; but do I get 1 damage or the amount of damage I rolled?

If this is a joke it's not funny ... :-)

If not, then like the card says, you get one damage and that's it

Twin Laser Turret makes two attacks. For each attack, you follow all* of the regular steps for an attack. So the attacker rolls dice, dice are modified, the defender rolls dice, those dice are modified, and then you compare results to see if the defender was hit.

If the defender is hit, the rest of the card's text kicks in: the defender suffers 1 damage, and then all dice results are canceled. That means when you get to the "deal damage" step, there are no [boom] or [kaboom] results left on the table, so the defender does not take any additional damage.

So the dice results you roll for TLT never deal damage directly, and are used solely to determine whether the attack hits. There are only two possible outcomes for each TLT attack: you hit (and the defender suffers one point of regular damage), or you miss (and the defender takes no damage).

[*Except that when you make the second attack, you skip the "declare target" step.]