The faucet in the bathroom sink is leaking. I've done some basica troubleshooting and it looks like it's from near the top top of the faucet, which I can't quite get a clean view on from the bottom. I know it's not from the pipes below, or the drainage. Basically I need to change the faucet.

I've never done it so I need some suggestions/instructions. I tried to turn off the main tap from the basement to stop the water. However, the water was still slowly dripping for a long while (over half an hour). I tried to disconnect the pipes underneath the sink to detach the faucet. But water was still kind of gushing out from it so I had to screw it back in and put the project on hold.

Question is, what have I done wrong, so far? And what should have I done otherwise? The main water tap was turned off and it's tight but still there was some small amount of water running. Should I be draining it for hours first? I couldn't find any other tap that controls the water for that bathroom specifically so I could only do the main tap. Need some hlep here. TIA