Is there any evidence for what has been mentioned about Surat Al-Falaq being a seeking of refuge from external evils, and Surat An-Nas being a seeking of refuge from internal evils, and that internal evils are more severe than external ones?

The aforementioned meaning is correct and is among the subtle points derived from contemplating the two "Mu'awwidhatayn" (Surahs Al-Falaq and An-Nas), around which exegetes have revolved. Surah Al-Falaq is for seeking refuge from the evil of what Allah created in general, then it specified certain evils such as the wicked person, the sorcerer, and the envious person. These are external evils stemming from the wrongdoing of others. As for Surah An-Nas, it came to seek refuge from a specific evil, which is the whisperer (Al-Waswas), and this is more specific than the general envious person and relates to internal defects whose origin is whispering (waswasah). Internal evils are more severe than external ones, which is why seeking refuge from the whisperer came with three attributes of Allah (the Lord, the King, the God), whereas seeking refuge from external evils came with one attribute (the Lord of daybreak).