3 sets of wires in outlet box, lights won’t turn off
You have hot wired your lights.

The light switch seems to be fed from the receptacle below it. (Cable C) So, the white wire on the switch should be re-identified as a hot wire with black, red or blue tape. It is now the hot feed to the switch. If this wire comes from the receptacle below it should also be re-identified there. The black wire in that same cable is the return leg from the switch. Your lights (Cable B) should be fed with the switch leg wire not the hot wire from the panel, and the neutral from Cable A.

In the receptacle box you should have a hot feed to the receptacle as one cable of the cables. (Cable A in your post.)

It sounds like the other cable (cable B) feeds your lights somehow. Strange way to wire it but from what you said that would be why your lights are now on all the time.

If you connected all the blacks and all the whites together in that box the lights would be on constant and the breaker would trip when you turn on the switch since it would now be a dead short. Since you didn't connect Cable C the switch does nothing at the moment.

Good luck and stay safe!