Mechanical cleaning is the only reliable way to clean a drain, period.

Edited to add suggestions and detail:

Most clogs are between the elbow and drain so something simple like the item below is more than enough to break free or pull out hair, and toothpaste, as well as anything else that might be stuck in drain or in the stopper parts. https://www.amazon.com/Grabber-Remover-Kitchen-Bathroom-Cleaning/dp/B07RMHNHYC/ref

In the case where the above is NOT enough, $30 gets you a cheap 25ft snake that is going to cover the bulk of other situations where you have a clog.
https://www.amazon.com/Plumbing-Adapter-Flexible-unblocking-Powered/dp/B0D46TQBQ9/ref

Some general basic notes: A food disposal is for the little bits that you cant scrape off into your compost or trash, ITS NOT A TRASH CAN. Please dont put stuff down the disposal that you could have removed from a plate or dish.

Fats and Grease go in the trash not down the drain. If you are pouring out your bacon grease or remaining cooking oil, or anything fatty it turns to a solid almost as soon as it hits the drain. Running hot water with it wont keep it liquid, and dish soap wont keep it emulsified. It will slowly build up in the pipes and you will get a fat burg/plug that causes a clog. Wipe fatty/greasy pans and dishes with a paper towel and throw it in the trash, for large quantities like Bacon grease you can use a can or jar that you store in the freezer until its full and then toss in the trash.