There is a really simple solution to this that does not involve burning a hole in your pipes with too much chemicals.
Look under your sink. You will see that the drain goes down and then turns back upward. This is called a sink trap. It is designed to trap objects so they don't flow into the sewer and get lost forever. Thus a ring or similar expensive item can be retrieved easily.
Get a pair of vice grips, channel locks, or pipe wrench. Put a bucket under the trap (it also traps water and is always full). Take off the two nuts that hold the trap on.
The trap will be filled with hair, toothpaste caps, bobby pins, and everything else imaginable. This is the reason for your slow drain. Clean the trap out using a wire hanger or whatever you can feed inside to get everything out of it. You probably want to wear some rubber gloves cause it WILL be disgusting.
Once clean, put the trap back on (don't forget to check for leaks and tighten as needed), and be prepared to be amazed as your water flows down the drain like you just installed a brand new sink.