I mean, the advanced docking computer does work sometimes; it's fine with planetary port landings and takeoffs, and it's fine with docking to mailslot stations. It will dock with rescue ships and fleet carriers, but it suffers overshoot and takes longer than going in manually. It really struggles with exiting mailslots, where it may bump the wall and take its own shield out, possibly destroy itself if left alone. This is mostly only a problem if it has to queue to leave; playing Solo in quiet stations it's ok, but Open or Mobius PvE at the CG station will likely leave you nose-to-paint.I'd thought of installing an advanced docking computer on it and not the standard one I currently have in all my ships, but sounds like that's not a great idea from what you said...
You can however disable auto launch in the ship menu without affecting the docking features, which gets rid of the only actually dangerous problem. For rescue ships, you can approach the pad under your own control and just zero the throttle when you are comfortably close and slow. It manages at shorter distances when it isn't tempted to go full throttle.
Personally, I would fit the advanced computer and hope it gets fixed at some point. I've noticed there's another thread that seems to be gaining traction, and the lateral thrust issues on large ships is one of the main issues being talked about. With luck, FDev will have to acknowledge and fix it; or at least, tune the autopilot to understand that the thrusters are weaker than it currently thinks they are.
Two hours of exploring and I haven't found a single bio yet
Type F, G and K stars are your best bet here. If you're already sticking to those then you're basically just unlucky