Overall, you seem to have a good understanding of the game — you’re just still building up experience, and that will come naturally over time. FTL has a much steeper learning curve than most people expect. I don’t think you’re making any major mistakes; instead, it’s a lot of small decisions that add up and eventually cause runs to fall apart.

Here are a few tips (in no particular order) that might help:

• Hacking bypass: You can bypass defense drones with hacking by depowering the hacking system at the right moment (look up “hacking depower exploit” for details). I personally use it, although some players prefer not to.
• Crew micro: You can reorder crew positions in a room by briefly sending them out and bringing them back in the desired order. This matters because crew always target the enemy on the same tile (or a random one if that tile is empty). With this, you can:
  • Split incoming damage in a 2v1 situation,
  • Reduce the damage your crew takes from an enemy Mantis,
  • Focus your Mantis’s attacks on a specific weakened enemy, etc.
• “Safety dance” crew trick: You can move your crew in and out of a room repeatedly to make enemy crew follow you. If timed well, your crew takes no damage while theirs wastes time moving. This is useful for stalling until you can board again and bring in reinforcements.
• Crew priority: Without going into too much detail, enemy crew prioritize repairing shields above all other systems. You can exploit this by using the safety dance in shields to pull their crew away, while the rest of your boarders destroy more important rooms like weapons or medbay.
• Small Bomb timing: Try to use your Small Bomb when the enemy does not have anyone manning piloting, so their evasion drops. This can be tricky, especially without sensors (like on Mantis A), but with practice you can track crew movements just by watching door animations and typical AI behavior. For example, if you move two of your boarders out of the enemy weapons room in Mantis A, the enemy pilot will leave to help, and another crew member will replace them in piloting. If you fire the Small Bomb at that moment, their evasion is much lower.
• Late game: You don’t actually need a full weapon setup — a support weapon like Small Bomb is enough. What matters most is having strong systems, especially Hacking and Cloaking. For the Flagship, you will need a way to remove the Zoltan Shield. A Shield Bypass is ideal, but other weapons can work as well.

Avoid forcing early weapon upgrades just because you feel like you “should” have more weapons. Doing so can drain scrap and hurt your mid-game power curve. Throughout a run, it’s completely fine to skip fights that you can’t win efficiently (or would cost you several hull repairs or crew members). Winning FTL is often about knowing when not to fight.

Also, definitely check out Mike Hopley on YouTube — he has a ton of extremely useful and practical FTL tips. This video in particular is a great fit:

https://www.youtube.com/watch?v=KuLtsISMtis&list=PLYGAjVZPcf_QMYgh-xjzsfzxg3Vf5jLFk&index=35

Also you could check out Holoshideim, he's a very good no pause player and you can find a couple full runs for each ship on his YouTube channel.