Answer
✅There is no prohibition against performing ablution with a wet face and hands, and in addition, the following narrations implicitly indicate that there is no need for the hands and face to be dry for ablution:
1 قَالَ أَبُو جَعْفَرٍ تَابِعْ بَيْنَ الْوُضُوءِ كَمَا قَالَ اللَّهُ عَزَّ وَ جَلَّ ابْدَأْ بِالْوَجْهِ ثُمَّ بِالْيَدَيْنِ ثُمَّ امْسَحِ الرَّأْسَ و الرِّجْلَيْنِ وَ لَا تُقَدِّمَنَّ شَيْئاً بَيْنَ يَدَيْ شَيْ ءٍ تُخَالِفْ مَا أُمِرْتَ بِهِ فَإِنْ غَسَلْتَ الذِّرَاعَ قَبْلَ الْوَجْهِ فَابْدَأْ بِالْوَجْهِ و أَعِدْ عَلَى الذِّرَاعِ وَ إِنْ مَسَحْتَ الرِّجْلَ قَبْلَ الرَّأْسِ فَامْسَحْ عَلَى الرَّأْسِ قَبْلَ الرِّجْلِ ثُمَّ أَعِدْ عَلَى الرِّجْلِ ابْدَأْ بِمَا بَدَأَ اللَّهُ عَزَّ وَ جَلَّ بِهِ
📚Kafi, Vol. 3, p. 34
Imam Baqir (peace be upon him) said: Perform ablution successively as Allah, the Exalted and Glorified, has commanded, begin with the face and then the two hands and then wipe your head and your two feet and do not put one thing before another in a way that you oppose what you have been commanded, so if you wash the hand before the face, then wash the face and repeat the washing of the hand, and if you wipe the foot before the head, then wipe the head before the foot and then repeat the wiping of the foot, (and) begin with what Allah has begun with.
2 عَنْ أَبِي عَبْدِ اللَّهِ فِي الرَّجُلِ يَتَوَضَّأُ فَيَبْدَأُ بِالشِّمَالِ قَبْلَ الْيَمِينِ قَالَ يَغْسِلُ الْيَمِينَ وَ يُعِيدُ الْيَسَار
📚Tahdhib, Vol. 1, p. 97
Imam Sadiq (peace be upon him) said about a person who performs ablution and begins with the left hand before the right hand: He washes the right hand and repeats the washing of the left hand.
3 عَنْ أَبِي عَبْدِ اللَّهِ.... قَالَ وَ إِنْ نَسِيَ شَيْئاً مِنَ الْوُضُوءِ الْمَفْرُوضِ فَعَلَيْهِ أَنْ يَبْدَأَ بِمَا نَسِيَ وَ يُعِيدَ مَا بَقِيَ لِتَمَامِ الْوُضُوء
📚Tahdhib, Vol. 1, p. 99
Imam Sadiq (peace be upon him) said in a hadith: If he forgets something from the obligations of ablution, it is his duty to begin with what he has forgotten and then repeat the rest of it to complete the ablution.
4 علي بن جَعْفَرٍ عن أَخِيهِ مُوسَى بْنِ جَعْفَرٍ قَالَ: سَأَلْتُهُ عَنْ رَجُلٍ تَوَضَّأَ فَغَسَل يَسَارَهُ قَبْلَ يَمِينِهِ كَيْف يَصْنَعُ قال يُعِيدُ الْوُضُوءَ مِنْ حَيْثُ أَخْطَأَ يَغْسلُ يَمِينَهُ ثُمَّ يَسَارَهُ ثُمَّ يَمْسَحُ رَأْسَهُ و رِجْلَيْهِ
📚Qurb al-Isnad, p. 83
Ali ibn Ja'far said: I asked Imam Kadhim (peace be upon him): A person performs ablution and washes his left hand before his right hand, what should he do? He said: He repeats the ablution from where he made the mistake: He washes the right hand and then washes the left hand and then wipes his head and his two feet.
5 عن زُرَارَة قَال: قَال لِي لَوْ أَنَّكَ تَوَضَّأْتَ فَجَعَلْتَ مَسْحَ الرِّجْلَيْنِ غَسْلًا ثُمَّ أَضْمَرْتَ أَنَّ ذَلِك مِنَ الْمَفْرُوضِ لَمْ يَكُنْ ذَلِكَ بِوُضُوءٍ ثُمَّ قَال ابْدَأْ بِالْمَسْحِ عَلَى الرِّجْلَيْنِ فَإِنْ بدَا لَكَ غَسْلٌ فَغَسَلْتَهُ فَامْسَحْ بَعْدَهُ لِيَكُون آخِرُ ذَلِكَ الْمَفْرُوض
📚Tahdhib, Vol. 1, p. 65
Zurarah said: The Imam said to me: If you perform ablution and instead of wiping, you wash your two feet and it is in your mind that this is the obligation, this is not a valid ablution, then he said: Begin with wiping the two feet, so if (for some reason) washing the feet came to your mind and you did it and you washed them, wipe them after washing so that this last wiping is for the obligation of ablution.
✅As can be seen in the above narrations, a person who has changed the order of ablution is obligated to go back to the place of the mistake and perform the ablution again until the end, and there is no mention of the need to dry the limbs, but rather, the washing of those limbs is simply repeated.
6 خَطَبَ أَمِيرُ الْمُؤْمِنِين فَقَالَ: إِنَّ اللَّهَ حَدَّ حُدُوداً فلَا تَعْتَدُوهَا وَ فَرَضَ فَرَائِضَ فَلَا تَنْقُصُوهَا و سَكَتَ عَنْ أَشْيَاءَ لَمْ يَسْكُتْ عَنْهَا نِسْيَاناً لَها فَلَا تَكَلَّفُوهَا رَحْمَةً مِنَ اللَّه لَكُم فَاقْبَلُوهَا
📚Faqih, Vol. 4, p. 74
Amir al-Mu'minin Ali (peace be upon him) said in a sermon: Indeed, Allah, the Blessed and Exalted, has specified limits, so do not transgress them, and He has established obligations, so do not diminish them (do not violate them), and He has remained silent about some matters, and He has not remained silent out of forgetfulness, so do not make them obligatory for yourselves, this is a mercy from Allah for you, so accept it.
✅As is clear from the noble narration above and others like it, matters such as performing ablution with a wet face and hands, about which Allah, the Exalted and Glorified, has remained silent and has not placed an obligation on us regarding them, we should not unnecessarily burden ourselves with an obligation in this regard.
The manner of ablution