What's a permanent solution to the weird depression/dimple under the corners of my lips?

The best way to resolve this is to inject your own fat. Starting with fillers is not a good choice since you are very young and you do not want to keep repeating this for the rest of your life. Also, it would make the judgement as to how much of the fat is still there very difficult, should you decide to have fat injection down the road. It creates a moving target.

Based on this photo it seem to be a loss of superficial, subcutaneous fat tissue (which comes with "aging"). Without in person evaluation, my first thought would be conservative correction with like tissue in form of fat grafts. Good luck.

You have two options for your depression. Surgical correction can be done by a expert plastic surgeon. Non surgical correction can be obtained by using a filler or fat. I would first attempt a non permanent filler and if you are happy, switch to a permanent filler. Good luck.

The very first question to ask is whether this is something you have had all your life or a new thing? If this is something you have had all your life and is attributed to the muscle structure of your face causing dimples it is not a good idea to begin with a long term filler or fat transfer. These types of initial approaches can cause irregularities or "lumpiness" over the natural facial dimples you have. Botox may be a good start but I agree, you would be doing it for awhile. A temporary filler is a good start as well to see whether a prolonged filler or fat transfer works in your case. It is ALWAYS about being conservative at first to see whether a result matches your expectations. Another quick note: It is NOT always about what your facial muscles and structure does in a relaxed position as this photo presents, rather, what happens when you smile and other facial expressions that allows for a true approach to improvement. Seek an experienced provider who can evaluate all those approaches for you.

These hollows are related to volume loss and in my clinic I would treat them with a flexible gel HA filler like Restylane Defyne. This is of course not a permanent treatment and the filler treatment would need to be repeated every 6-12 months.

The depression is from a retaining ligament called mandibular cutaneous ligament and not a muscle . Surgical repair would be the long term solution