I would treat full auto as an AoE, DC 8+ proficiency bonus+ dex modifer half damage on successful save. Base damage 3d8, burst fire mode -2 to hit 4d8 damage uses 3 shots.
Zat's maybe make them a dex save, you fail and you're unconscious, 2nd hit dead, 3rd hit disintegration. Give them low ammo capacity and can't be bought, you have to raid the Gou'ald.
Unfortunatrly, the specs you give for zat dont match the series.
Looking at their scuffed and bloodied idea, I would try the following
CORE mechanic: Exposed. Most "hits" by any weapon cause hp damage. If a character is "exposed" each weapon type has additional effects that apply. Exposed is a condition that equates to being open for the more serious effects of a "good hit" to apply.
A character is considered exposed in cases like:
The character is bloodied (below 50%hp)
The hit is a crit.
The character is surprised.
For zat, the exposed effect eould be a con save with success meaning you get one turn before going to unconscious. Failure means going to unconscious. (Or certain features of character might allow retaliation strikes)
For gunshots, you get saves vs "shock".
Auto fire should in some way make exposing more likely- higher crit chance maybe) especially when used to hose down the enemy.
This general model likely produces a combat style where surprise does wonders st dropping folks, the first flurry of straight up combat is survivable but you are well incentivized to find cover.
Just one spproach.