Is a make-up fast obligatory for someone who intended to break their fast while fasting, yet did not eat or drink due to obsessive doubts about their fast being invalid? And does forgetting to make the intention before dawn and remembering it after the Adhan affect the validity of the fast?

You must completely disregard obsessive thoughts and pay no attention to them. Allah is All-Compassionate, All-Merciful, and He does not burden His servants with more than they can bear. The default assumption is the validity of worship, and it is not invalidated except by certainty.

Swallowing saliva does not invalidate the fast, by scholarly consensus, and it is not necessary to wipe the lips. If a small amount of food trapped between the teeth reaches the stomach unintentionally, there is no harm, and some scholars have even permitted swallowing a small amount of it.

Intentionally breaking the fast invalidates it, even if no eating or drinking accompanies it. However, what occurs under the pressure of obsessive thoughts is excused. Therefore, the intention to break the fast does not invalidate it unless one actually commits an act that breaks the fast.

As for forming the intention overnight (for the next day's fast), merely thinking that you will fast tomorrow constitutes the obligatory intention. There is no need to utter it verbally or repeat it.