Oh yeah, it's an eternal problem, my friend and I'm sure that each of us has come across this. In such cases, there are several solutions, draining in the sink works on the principle of a sump and you need to disassemble the drainage device under the sink and clean it, there are also special chemicals that dissolve the blockages and they are sold in stores and finally, if nothing helps I use a small plumbing cable (photo at the bottom), which is used by plumbers, it must be gradually pushed into the pipe and it breaks the jam; it is a simple, but 100% tool for cleaning pipes. Thank you Mark and I hope that I somehow helped in your problem.
I called a plumber before years ago for a blockage in the garage. He had one of those things that rolled out of his van that worked on water pressure. He said it was so powerful it could drill a hole in your hand. It did the trick.