it's pure luck. You should still try for the lesser rooms after finding the Sack, and just get better at not losing health to them :)

But I would also skip bossfights after the fountain. That's why I start with the harder rooms.

That's normal. I've been getting it on my second chamber a lot as well. You'll be thankful in the future as once you reach end-game (high-heat runs) and find the game a lot more easy due to your experience, you will appreciate the time saved by getting it straight away (first chamber will never have the sack so the earliest you can get it is at the second one).

You can play it to your favor I guess, go for dangerous labyrinths first, get the sack out of the 2nd one, other labyrinths will be less dangerous.
I do not think I've ever got that sack later than 3rd labyrinth, so it's no point in being cautious. I do not lose too much hp in those anyway now, as long as I have a dash damage. Just evaluate if it's profitable to go to the left-over labyrinths, if you feel like you are prepared enough - just got kick that old man's ass with what you already have. )