I recently just made the jump to normal and it is considerably more difficult. I can't get past maybe the 4th jump? Just right now, I was maybe halfway and for whatever reason my ship was running out of oxygen, which was odd, I had no damage to my o2, no crack in the hull nor were my doors open. I was a little annoyed, but I just can't stay mad at the game, I am sure I made an oversight at some point.
Take your time in the first two sectors in Normal. Try to hit as many points as possible before moving on so you can load up on scrap and upgrade. I try to level up my shields at least once by sector 3, engines maybe 2-3 levels. And PAUSE THE GAME TO PLAN OUR MOVES. There's no shame in doing so. To save scrap, don't ever fully repair at stores, maybe just to 70-75%. There's a random event where you can get rewarded with hull repairs.
Must have augments for me include Long Range Scanners, Scrap Recovery Arm, Drone Recovery Arm, and the Weapon/Shield Boosters. Ditch the Scrap Recovery when you've upgraded enough and use a different Augmentation. Drone Recovery essentially gives you unlimited drones, provided you have at least 1 in stock.
As far as weapons go, Burst Laser Mk II is the best in the game (damage/power req. ratio). Ion Blast Mk I is good, Mk II is beastly. Keep a missile launcher around if you find yourself unable to pierce shields (you can just keep in storage, don't have to equip it if you have weapons you're doing well with). But with the randomness of the game, you might not see these in a run. so I don't plan around them. But if I see them available, I try to make it a priority to get them.
Oxygen dropping? Are you sure you had Life Support powered on? Sometimes it gets shut off in a random event and I forget to turn it back on later.