I wrote a post about unclogging drains that might help - https://chascrazycreations.com/unclog-drains-toilets-with-out-plunging/
Baking soda, vinegar and hot water.
Buy Sink and Plug Unlocker from the Supermarket or invest in a Plumbers
Plunger............I assume you have tried filling the sink with water and plunging a container up and down to create a Vaccum . Best wishes
Green Gobbler sold in many stores.
Even though we have an auger, most commonly we find that our drains need to be physically fished out. Either using an unbent hanger as a refashioned hook or plastic barbed tool from the dollar store to explore the drain obstruction.
Fishing out the blockage with that in -our experience— to partially involve hair seems to most readily alleviate stoppage in our sink and shower drains.
If there is a problem with a tub drain-There are some retrofit devices out there and that could be helpful if you get your drain clean consider the little mushroom cap for collecting the hair. There is also a more refined looking chrome for one but if you look at customer reviews and Q&A’s it seems like this mushroom one has a higher review. You may consider having a look at these existing devices to see if that will help the problem from from reoccurring.
Teresa....There is a product called Liquid fire. It can be purchased on Ebay.