Dead drunk is just the highest level of drunkeness. She's is more resistant to alcohol with higher stamina level, but especially the later unlocked harder drinks pack a punch. An indicator for drunkeness is ofc her face, but also the time it takes her to do tasks. Each level of drunkeness increases the time by 1s. Other influences are arousal, wet, horny and titles. So just remember the base time for an action and figure out how many seconds are due to alcohol (e.g. a 10s action takes an aroused and wet Karryn 15s, now you know she's in the 'drunk' state).I think dead drunk means maxing out the drunkenness score the game uses behind the scenes. Somebody on the discord laid it out, IIRC, the drunken gangbang scene happens as a combination of drunkenness and cock desire, so the trick is to keep cock desire low while drinking to get the score higher. Apparently, if you can pull it off, they don't dump out the beer from the mug and you can have her continue to drink even in the gangbang. I'm guessing this is something that's much easier to do in the early game before your slut level is too high.
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Now, regarding the sex scene and the beer:
She needs to be at least 'tipsy' (i.e. just a bit drunk) or it cannot fire. Also she must fulfill all requirements for handjob or blow job or vag sex or anal sex or is dead drunk. So keeping cock desire low keeps the scene from happening for long. (Earlier versions lowered all requirements with higher drunkeness but not the current version).
The beer gets replaced once she fulfills all requirements to swallow cum. This usually requires at least 100 cock and mouth desire. It's entirely possible (I tested that for a tweak) to keep her beer refilled by keeping cock desire low enough even though she wasn't dead drunk.
Why is it still easier starting the scene dead drunk? Because you can trigger the scene with very low cock desire if she's dead drunk and then in the scene nobody can engage in hj, vag or anal. So they all just jerk off and don't do much to increase her cock desire past 100. Keeping mouth desire low enough to stop bj prevents the swallow requirement, too.